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FOUNDATIONS  OF 
FORMAL  LOGIC 


BY 


HENRY  BRADFORD  SMITH 


PHILADELPHIA 
PRESS  OF  THE  UNIVERSITY  OF  PENNSYLVANIA 

1922 


144  0 


IB  1 


3C 


i 


PREFACE 


The  edition  of  his  Primer  of  Logic  being  exhausted,  the  writer 
is  forced  to  prepare  another  outline  for  classroom  use.  Only 
parts  of  that  text,  however,  have  been  retained  and  these 
*V  have  received  a  methodical  rearrangement  and  expansion.  Two 
*  chapters  of  the  work,  Non-Aristotelian  Logic,  appear  again  but 
modified  in  detail.  To  these  have  been  added  historical  notes 
and  citations  and  three  new  final  chapters  and  the  text  has  been 
illustrated  by  a  number  of  diagrams. 

The  writer  has  again  to  express  his  indebtedness  to  Professor 
Singer  for  his  introduction  to  the  method  which  is  here  employed. 
This  indebtedness  is  to  be  referred  not  only  to  the  Syllabus  of  his 
lectures  (reprinted  pp.  46-52  in  the  writer's  Letters  on  Logic)  but 
tj        also  to  many  hints  thrown  out  in  private  discussion, 
v  The  present  work,  as  its  title  suggests,  does  not  pretend  that 

its  system  is  completely  developed.  Its  chief  concern  is  with  the 
foundations,  upon  which  a  theory  may  be  built.  Its  solutions 
would  have  been  carried  farther,  if  the  writer  could  have  found 
his  way  through  to  the  end. 

H.  B.  S. 


TABLE  OF  CONTENTS 
CHAPTER  I 

PAGES 

§  1-6     Forms  of  proposition  recognized  by  the  logician.  .      7-13 

CHAPTER   II 

§  7-11     Relations  of  "better"  and  "worse."     Immediate 

inference 14-18 

CHAPTER  III 
§  12-16     Moods  of  the  syllogism 19-25 

CHAPTER   IV 
§  17-22     General  solution  of  the  sorites 26-32 

CHAPTER  V 
§  23-31     Foundations  of  the  calculus 33-40 

CHAPTER  VI 
§  32-35     General  solution  of  the  syllogism 41-44 

CHAPTER  VII 
§  36-40     On  certain  supposed  fallacies 45-49 

CHAPTER  VIII 
§  41-42    Alternative  systems.     Non-Aristotelian  logic.  .  .  .   50-51 

CHAPTER   IX 
§  43-46    The  classical  system  and  its  verification 52-56 


CHAPTER  I 

§  1 .  The  problem  of  a  deductive  science  is  to  define  its  elements, 
the  objects  of  which  it  treats,  by  an  enumeration  of  their  formal 
properties.  These  properties  are  to  be  found  within  the  system 
of  which  these  objects  are  the  parts.  The  task  of  logic1  is,  then, 
to  develop  its  own  system  by  constructing  all  the  true  and  all  the 
untrue  propositions  into  which  its  elements  enter  exclusively. 

According  to  Schroder,  if  a,  b,  c  ...  be  a  set  of  indefinables  and 
R  a  relation,  then  a,  b,  c  .  .  .  R  are  said  to  form  a  system,  provided 
P  R  q  stand  for  an  assertion  that  must  be  either  true  or  false  and  can  not 
be  both  true  and  false, — p  and  q  representing  any  members  of  the  set,  a, 
b,  c  .  .  .  (See  Schroder's  Abriss  der  Algebra  der  Logik,  by  Dr.  Eugen 
Muller,  Leipzig,  1909,  p.  18.) 

§  2.  We  shall  begin  with  an  enumeration  of  the  forms  with  which 
logic  deals. 

The  Categorical  forms  are  composed  of  terms  and  relationships 
and  are  represented  by  the  following  propositional/wHc/tows: 

a(ab)  =  All  a  is  all  b, 
(l(ab)  =  Some  a  is  some  b, 
y(ab)  =  All  a  is  some  b, 
e(ab)  =  No  a  is  b. 

These  are  called  propositional  functions  because  each  one  is 
true  only  for  some,  not  for  all  meanings  of  a  and  6;  because  their 
truth  or  untruth  depends  upon  the  meaning  of  a  and  b.  When- 
ever it  is  desired  to  designate  indifferently  any  one  of  these  (i.  e., 
to  leave  it  unsettled  which  one  is  meant)  the  notation  x(ab), 
y(ab),  etc.,  will  be  employed.  In  the  proposition  x{ab)  the  terms 
are  the  subject  a  and  the  predicate  b  and  the  term  order  is  the  order 
subject-predicate.  Whenever  the  term  order  is  unsettled  a 
comma  will  appear  between  the  terms.  Thus  x(a,  b)  may  mean 
either  x(ab)  or  x{ba).  The  relationships  in  the  categorical  forms 
are  the  copula  is  and  the  adjectives  of  quantity,  all,  some  and  no. 

1  Logic  is  from  XoytKij,  an  adjective  with  a  substantive  understood,  and 
\oyiKrj  is  from  Xo'yos,  which  denoted  both  the  thought  and  its  expression 
(ratio  and  oratio).  This  ambiguity  passed  into  the  meaning  of  the  derivative 
XoytKrj  and  led  indirectly  to  a  dispute  as  to  whether  Logic  deals  with  the 
laws  of  thought  or  with  the  laws  of  the  expression  of  thought. 


8  Foundations  of  Formal  Logic 

The  word  some  is  ambiguous  in  ordinary  speech,  meaning  either 
some  at  least,  possibly  all  or  some  at  most,  some  at  least,  not  all.  It 
is  this  latter  interpretation,  which  we  shall  assume  to  be  forced 
upon  us  by  the  propositions  which  we  say  shall  be  true  or  untrue 
in  our  science. 

In  1846  Sir  William  Hamilton  published  the  prospectus  of  an  essay 
on  a  New  Analytic  of  Logical  Forms,  which  revived  the  question  as  to 
whether  or  not  the  quantity  of  the  predicate  should  be  made  explicit. 
The  chief  difficulties  of  his  system  result  from  the  ambiguity  of  the 
meaning  of  some,  from  the  effort  to  make  every  form  of  categorical 
expression  simply  convertible  and  from  the  seemingly  curious  effort  to 
establish  an  order  of  belter  and  worse  among  the  relations  connecting 
subject  and  predicate.  Four  of  Hamilton's  eight  forms  are  redundant. 
Those  that  are  essential  are  represented  here  by  the  letters,  a,  /3,  y  and  e. 

§  3.  The  terms,  a  and  b,  stand  for  classes,  for  a  group  of  objects 
conceived  by  the  aid  of  a  common  property.  Every  substantive 
in  the  language  is  the  symbol  for  such  a  group.  There  is  a  certain 
analogy  between  the  manner  in  which  closed  areas  overlap  and 
the  manner  in  which  classes  overlap,  which  was  first  pointed  out 
by  the  mathematician  Euler.  This  analogy  often  breaks  down, 
as  might  be  expected,  for  some  limiting  case  and  has  perhaps  led 
the  logician  astray  as  often  as  it  has  aided  him.  It  will  prove  to 
be  invaluable,  however,  in  enabling  us  to  attach  a  preliminary 
meaning  to  our  symbols  of  relationship. 

"As  a  general  notion  contains  an  infinite  number  of  individual  objects, 
we  may  consider  it  a  space  in  which  they  are  all  contained.  Thus  for  the 
notion  of  man  we  form  a  space,  in  which  we  conceive  all  men  to  be  com- 
prehended. For  the  notion  of  mortal  we  form  another  in  which  we  con- 
ceive everything  mortal  to  be  comprehended.  And  when  I  affirm  all 
men  are  mortal  it  is  the  same  thing  with  affirming  that  the  first  figure  is 
contained  in  the  second.  .  .  .  These  circles,  or  rather  these  spaces,  for 
it  is  of  no  importance  of  what  figure  they  are,  are  extremely  commodious 
for  facilitating  our  reflections  on  this  subject,  and  for  unfolding  all  the 
boasted  mysteries  of  logic,  which  that  art  finds  it  so  difficult  to  explain; 
whereas  by  means  of  these  signs  the  whole  is  rendered  sensible  to  the 
eye."  (Letters  of  Euhr  addressed  to  a  German  Princess,  by  David  Brewster, 
New  York,  1846,  Vol.  I,  pp.  339-341.) 

The  diagrammatic  representation  of  the  categorical  forms  is 
given  below. 


Foundations  of  Formal  Logic 


All  a  is  all  b 


Some  a  is  some  b 


All  a  is  some  b 


All  b  is  some  a 


No  a  is  b 


Assuming  now  that  these  forms  exhaust  all  of  the  modes  in 
which  two  closed  areas  may  overlap,  it  will  seem  natural  to 
declare,  on  the  ground  of  our  analogy,  that  any  two  classes, 
a  and  b,  must  be  related  in  one,  and  can  not  be  related  in  more 
than  one,  of  these  five  ways.  The  assertion  that  a  and  b  must 
realize  one  of  these  possibilities  we  shall  term  the  propositional 
universe.  The  assertion  that  a  and  b  must  realize  more  than 
one  of  these  possibilities  we  shall  term  the  propositional  null. 
The  former  assertion  will  then  appear  to  be  true  for  all  meanings 
of  the  terms  and  the  latter  assertion  will  appear  to  be  untrue  for 
all  meanings  of  the  terms. 

§  4.  The  remaining  forms  of  proposition,  which  are  recognized 
by  the  logician,  are: 


10  Foundations  of  Formal  Logic 

The  Hypothetical  form, 

if  x (is  true)  then  y(is  true), 

=  x  implies  y  =  x  Z  y, 

x  does  not  imply  y  =  (x  Z  y)'\ 

The  Conjunctive  form, 

#(is  true)  and  y(is  true), 
=  xy\ 

The  Disjunctive  form, 

either  #(is  true)  or  >>(is  true), 
=  x  +  y. 

Here  *  and  y  in  turn  stand  for  any  sort  of  proposition.  If  they 
happen  to  be  categorical  forms,  then  we  should  replace  the  abbre- 
viations above  by  the  more  definite  notation,  x(a,  b)  Z  y(a,  b), 
x(a,  b)y(a,  b),  x(a,  b)  +  y(a,  b).  The  untruth  of  x  will  be  denoted 
by  x',  the  untruth  of  x'  by  x" . 

The  relations  of  inclusion  and  implication  are  usually  rendered  by 
Peano's  sign  of  a  flat  "C"  opening  to  the  left,  the  initial  letter  of  the 
word  contains;  or  by  Schroder's  half  bracket  opening  to  the  right  and 
drawn  through  an  equality  sign.  The  symbol  of  the  text  is  a  simplifica- 
tion of  the  one  employed  by  Schroder,  only  the  upper  half  being  retained. 

§  5.  The  word  "true"  never  connotes  possibly  true  or  true  in 
some  instances.  It  means  necessarily  true,  true  for  all  cases,  true 
for  all  meanings  of  the  terms.  The  following  propositions  may  be 
verified  at  once  "empirically"  by  the  aid  of  Euler's  diagrams. 
If  the  student  will  examine  each  illustration  with  care,  he  will 
begin  to  attach  a  meaning  to  the  symbolism.  Let  him  observe 
that  the  figure  in  each  case  represents  the  part  to  the  left  of  the 
implication  sign  as  true  and  the  part  to  the  right  of  the  implica- 
tion sign  as  true  at  the  same  time. 


a(ab) 


Foundations  of  Formal  Logic 


11 


y(ab)  0(cb)  Z  y\ca) 


"Inference  does  not  give  us  more  than  there  was  before:  but  it  makes 
us  see  more  than  we  saw  before.  .  .  .  The  homely  truth  that  no  more 
can  come  out  than  was  in,  .  .  .  has  been  applied  to  logic,  and  even 
to  mathematics,  in  depreciation  of  their  rank  as  branches  of  knowl- 
edge. .  .  .  and  hence  some  have  spoken  as  if  in  studying  how  to  draw 
the  conclusion,  we  are  studying  to  know  what  we  knew  before.   .    .    . 

"The  study  of  logic,  .  .  .  considered  relatively  to  human  knowl- 
edge, stands  in  as  low  a  place  as  that  of  the  humble  rules  of  arithmetic, 
with  reference  to  the  vast  extent  of  mathematics  and  their  physical 
applications.  Neither  is  the  less  important  for  its  lowliness:  but  it  is 
not  every  one  who  can  see  that."  (De  Morgan,  Formal  Logic,  London, 
1847,  pp.  44-46.) 

The  word  "untrue"  means  not  necessarily  true,  not  true  in  all 
instances.  Accordingly,  in  order  to  establish  the  untruth  of  a 
given  proposition,  it  will  be  enough  to  point  to  a  single  instance 
of  its  being  untrue.  The  diagrams  which  follow  will  establish 
"empirically"  the  untruth  of  the  proposition  to  which  each  one 
corresponds.  The  fact  to  be  noticed  here  is  that  the  figure  in 
each  case  represents  the  part  to  the  left  of  the  implication  sign 
as  true  and  the  part  to  the  right  of  the  implication  sign  as  false 
at  the  same  time. 


y(ab)  y(bc)  Z  y(ca) 


y(bc)  a(ba)  Z  y\ac) 


12 


Foundations  of  Formal  Logic 


§  6.  If  we  refer  again  to  the  five  possible  modes  of  representing 
the  formal  relationship  of  a  to  b,  viz., 


it  will  be  easy  to  understand  what  is  contained  in  the  statement, 
that  x(ab)  is  false.  This  will  be  the  same  as  to  assert  that  the 
possibility  for  which  x(ab)  stands,  is  to  be  left  out  of  the  universe 
and  the  sum  of  the  remaining  possibilities  asserted  to  be  true. 
The  prime  over  x(ab)  may  be  conceived  as  an  operator  which 
strikes  out  the  corresponding  diagram.  Thus  e'(ab)  will  be  repre- 
sented precisely  by  the  disjunction : 


or 


Whenever  the  conjunction  of  two  or  more  propositions  stands 
for  an  impossibility,  their  product  is  said  to  be  null.  Thus,  the 
product  of  a  and  /3  is  null,  but  the  product  of  a'  and  /3'  is  not  null. 

Whenever  the  disjunction  of  two  or  more  propositions  exhausts 
all  of  the  possibilities  that  there  are,  their  sum  is  said  to  be  universe. 
Thus,  the  sum  of  a  and  /3'  is  universe,  but  the  sum  of  a  and  /3  is 
not  universe. 

The  following  definitions  are  of  importance  for  our  subsequent 
theory : 

(1)  If  the  logical  sum  of  two  categorical  forms  is  the  universe 
and  their  logical  product  is  null,  the  two  are  said  to  be  contra- 
dictory forms. 

(2)  If  the  logical  sum  of  two  categorical  forms  does  not 
exhaust  the  universe  but  their  logical  product  is  null,  the  two 
are  said  to  be  contrary  forms. 


Foundations  of  Formal  Logic  13 

(3)  If  the  logical  sum  of  two  categorical  forms  is  the  universe 
but  their  logical  product  is  not  null,  the  two  are  said  to  be 
subcontrary  forms. 

(4)  If  the  logical  sum  of  two  categorical  forms  does  not 
exhaust  the  universe  and  their  logical  product  is  not  null,  the 
two  are  said  to  be  subaltemate  forms. 

Suppose,  now,  that  k(ab)  and  w(a6)  designate  any  one  of  the 
unprimed  categorical  forms  but  cannot  designate  the  same  form, 
y(ab)  and  y(ba)  being  considered  distinct,  then  the  following 
statements  may  be  verified  at  once  as  the  result  of  a  complete 
induction  of  all  of  the  possibilities : 

(1)  k(ab)  and  k'(ab)  are  contradictory; 

(2)  k(ab)  and  w(a6)  are  contrary; 

(3)  k'(ab)  and  w'(a6)  are  subcontrary ; 

(4)  k(ab)  and  w'(ab)  are  subaltern. 


CHAPTER   II 

§  7.  At  this  point  in  our  theory  we  shall  introduce  certain  rela- 
tions, which  will  be  termed  the  distinctions  of  better  and  worse, 
following  a  suggestion  of  Sir  William  Hamilton's  making  (Lectures 
on  Logic,  Appendix,  p.  536).  It  will  turn  out  in  the  sequel  that 
these  assumed  indefinables  may  be  dispensed  with.  A  study  of 
them,  however,  has  an  historical  interest  and  will  serve  the  purpose 
of  making  meaningful  the  process  of  deduction.  They  fulfil  the 
spirit  rather  than  the  letter  of  Hamilton's  thought. 

Let  us  begin  by  inventing  symbols  to  denote  our  distinctions,  i.  e., 

x  I  y  =  x  is  singly  worse  than  y, 
%  II  y  =  oc  is  doubly  worse  than  y, 
x  III  7  =  x  is  trebly  worse  than  y 

and  let  us  add  the  following : 

Definition. — In  the  propositions,  x  /  y,  x  //  y  and  x  ///  y,  the 
relation  connecting  x  and  y  is  known  as  the  worse-relation. 

Definition. — Singly  worse,  doubly  worse  and  trebly  worse  are 
known  as  the  three  degrees  of  the  worse-relation. 

Let  us  suppose  now  that  x  and  y  can  represent  only  the  categor- 
ical forms  and  let  us  imagine  x  and  y  to  take  on  in  all  possible  ways 
the  values,  a,  p,  y  and  e.  There  will  then  be  sixteen  possible 
propositions  of  each  type,  x  /  y,  x  H  y  and  x  ///  y,  obtained  by 
permuting  the  letters  two  at  a  time  and  by  taking  each  letter  once 
with  itself.  If  the  symbol  of  relationship  be  omitted  and  each 
proposition  be  exhibited  as  a  simple  combination  of  the  two  letters, 
then  each  set  of  sixteen  would  appear  thus : 


aa 

Pa 

7a 

ca 

a/3 

08 

70 

e/8 

ay 

py 

77 

€7 

at 

Qt 

7« 

tt 

All  of  the  propositions  of  this  set  taken  together  are  known  as 
the  array  of  whichever  type,  %  /  y,  x  1 1  y  or  x  l/l  y,  they  particular- 
ize. Each  proposition  of  the  array  is  known  as  a  mood  of  the  array. 
True  moods  of  the  array  are  known  as  valid  moods  of  the  array. 
The  remaining  moods  are  known  as  invalid  moods  of  the  array. 

14 


Foundations  of  Formal  Logic  15 

§  8.  The  three  arrays,  once  the  valid  and  invalid  moods  have 
been  established,  constitute  a  species  of  definition  of  the  three 
degrees  of  the  worse-relation,  for  they  exhaust  all  of  the  proposi- 
tions into  which  this  relationship  may  enter.  We  proceed  to 
formulate  certain  rules  and  to  assume  certain  postulates  by  the 
aid  of  which  all  of  the  moods,  valid  and  invalid,  may  be  deduced. 

Definition  1. — In  the  propositions,  x  /  y,  x  //  y  and  x  ///  y,  x  is 
called  the  inferior  and  y  is  called  the  superior  form. 

Definition  2. — Trebly  worse  is  singly  worse  than  doubly  worse 
and  doubly  worse  is  singly  worse  than  singly  worse. 

Rule  1. — If  in  any  valid  mood  inferior  form  and  worse-relation 
be  each  made  singly  worse,  a  valid  mood  will  result. 

Postulate  1. — /3  /  a  is  a  valid  mood. 

Postulate  2. — 7  /  /3  is  a  valid  mood. 

Postulate  3. — c  /  7  is  a  valid  mood. 
Applying  rule  1  to  postulate  1 ,  we  obtain  at  once 

Theorem  1. — 7  //  a  is  a  valid  mood. 

(Postulate  2  and  Definitions  1  and  2). 
Applying  rule  1  to  theorem  1 ,  we  obtain  at  once 

Theorem  2. — e  ///  a  is  a  valid  mood. 

(Postulate  3  and  Definitions  1  and  2.) 
Applying  rule  1  to  postulate  2,  we  obtain  at  once 

Theorem  3. — e  ///Ska  valid  mood. 

(Postulate  3  and  Definitions  1  and  2.) 

The  rules  and  postulates  for  the  deduction  of  the  invalid  moods 
are  given  below.  The  derivation  of  the  thirty-nine  theorems  is 
left  as  an  exercise  for  the  student. 

Definition  3. — If  x  is  worse  than  y,  then  y  is  said  to  be  better  than 
x  in  the  corresponding  degree. 

Rule  2. — If  in  any  invalid  mood  superior  and  inferior  form  be 
each  made  singly  better  or  each  made  singly  worse,  an  invalid 
mood  will  result. 

Rule  3. — If  any  invalid  mood  inferior  form  and  worse-relation 
be  each  made  singly  worse,  an  invalid  mood  will  result. 

Rule  4. — If  in  any  invalid  mood  superior  form  be  made  trebly 
worse,  an  invalid  mood  will  result. 

Postulate  4. — a  /  7  is  an  invalid  mood. 

Postulate  5. — 7  /  a  is  an  invalid  mood. 

Postulate  6. — e  /  a  is  an  invalid  mood. 

Theorems. — The  remaining  (39)  invalid  moods. 


16  Foundations  of  Formal  Logic 

The  "order"  of  better  and  worse  among  the  four  forms  being 
now  unambiguously  established  by  postulate  and  theorem,  we 
may  exhibit  our  result  schematically  thus : 

Best        a — 0 — 7 — e        Worst 

§  9.  In  the  hypothetical  forms,  x  Z  y  and  (x  Z  y)',  the  part  x 
to  the  left  of  the  implication  sign,  is  called  the  antecedent  and  the 
part  y  to  the  right  of  the  implication  sign  is  called  the  consequent. 

Here  x  and  y  may  stand  for  any  sort  of  proposition  but  if  each 
one  happens  to  represent  a  single  categorical  form,  we  should  then 
replace  x  Z.  y  and  (x  Z  y)'  by  the  more  definite  notation,  x(a,  b) 
Z.y(a,b)  and  [x(a,  b)  Z.y{a,b)\'.  Any  implication  of  this 
specific  type  is  known  as  immediate  inference. 

A  difference  between  two  forms  of  inference  which  is  dependent 
on  term-order  alone,  is  known  as  a  difference  of  figure.  Thus, 
x(a,  b)  Z  y(a,  b)  may  have  either  one  of  two  forms.  If  the  term- 
order  in  the  antecedent  is  the  same  as  the  term  order  in  the  conse- 
quent, i.  e.,  if  x(a,  b)  Z  y(a,  b)  be  written 

either  x{ab)  Z  y(ab), 
or        x(ba)  Z  y(ba), 

then  x(a,  b)  Z  y(a,  b)  is  said  to  be  expressed  in  the  first  figure  of 
immediate  inference.  If  the  term-order  in  the  antecedent  is  the 
reverse  of  the  term-order  in  the  consequent,  i.  e.,  if  x(a,  b)  Z  y(a,  b) 
be  written 

either  x(ab)  Z  y(ba), 

or        x(ba)  Z  y(ab), 

then  x(a,  b)  Z  y(a,b)  is  said  to  be  expressed  in  the  second  figure 
of  immediate  inference. 

Just  as  the  comma  between  the  terms  means  that  the  term- 
order  is  not  settled,  so  the  x  in  x(a,  b)  and  the  y  in  y(a,  b)  mean 
that  the  categorical  form  for  which  x(a,  b)  or  y(a,  b)  stands  is 
undetermined.  Suppose  now  that  we  should  particularize  x  and  y, 
that  is,  allow  them  to  take  on  in  every  possible  way  their  four 
specific  values,  a,  /3,  y  and  e.  There  will  evidently  result  sixteen 
distinct  moods  of  the  array  x(a,  b)  Z  y(a,  b)  in  each  one  of  the 
two  figures. 

§  10.  If  x(ab)  Z  x(ba)  is  a  valid  mood  of  immediate  inference, 
then  x{ab)  is  said  to  be  a  convertible  form.     The  operation  of  simple 


Foundations  of  Formal  Logic  17 

conversion  consists  in  the  interchange  of  subject  and  predicate. 
Referring  again  to  the  diagrams  of  the  last  chapter,  it  will  seem 
natural  to  attach  this  property  of  simple  convertibility  to  a,  /3  and  e. 
Employing  this  terminology  we  should  then  say  that  a,  /3  and  « 
are  convertible  forms  or  that  a,  0  and  e  are  simply  convertible.  In 
order  to  give  expression  to  this  fact,  let  the  following  assumptions 
be  granted: 

Postulate  1. — a(ab)  Z  a (6a)  is  a  valid  mood. 

Postulate  2. — (3(ab)  Z  j8(6a)  is  a  valid  mood. 

Postulate  3. — y(ab)  Z  y(ba)  is  an  invalid  mood. 

Postulate  4. — e(ab)  Z  c(6a)  is  a  valid  mood. 

We  shall  now  set  down  the  rules  for  the  deduction  of  the  valid 
moods  of  immediate  inference : 

Rule  1 . — If  in  any  valid  mood  the  subject  and  predicate  be  inter- 
changed in  a  form  that  is  simply  convertible,  a  valid  mood  will 
result. 

Rule  2. — If  in  any  valid  mood  of  the  first  figure  the  antecedent 
and  the  consequent  be  each  made  singly  worse,  a  valid  mood  will 
result. 

Theorem  1. — a(ab)  Z  a(ab)  is  a  valid  mood. 

(Postulate  1  and  Rule  1.) 
Theorem  2. — {l(ab)  Z  /3(o6)  is  a  valid  mood, 

(Theorem  1  and  Rule  2.) 
Theorem  3. — y(ab)  Z  y(ab)  is  a  valid  mood, 

(Theorem  2  and  Rule  2.) 
Theorem  4. — e(ab)  Z  e(ab)  is  a  valid  mood, 

(Theorem  3  and  Rule  2.) 

The  rules  and  postulates  for  the  deduction  of  the  invalid  moods 
are  given  below.  The  derivation  of  the  twenty-one  theorems  is 
left  as  an  exercise  for  the  student. 

Rule  1. — If  in  any  invalid  mood  the  subject  and  predicate  be 
interchanged  in  a  form  that  is  simply  convertible,  an  invalid  mood 
will  result. 

Rule  2. — If  in  any  invalid  mood  of  the  first  figure,  the  antecedent 
and  the  consequent  be  each  made  singly  worse,  an  invalid  mood 
will  result. 

Rule  3. — If  in  any  invalid  mood  of  the  first  figure  the  antecedent 
and  the  consequent  be  interchanged,  an  invalid  mood  will  result. 


18  Foundations  of  Formal  Logic 

Postulate  5. — a(ab)  Z  j3(ab)  is  an  invalid  mood. 
Postulate  6. — a{ab)  L  y(ab)  is  an  invalid  mood. 
Postulate  7. — a(ab)  Z   e(ab)  is  an  invalid  mood. 
Theorems. — The  other  (21)  invalid  moods. 

§  11.  We  may  also  formulate  rules  for  the  immediate  detection 
of  the  invalid  moods,  as  follows: 

Rule  1. — If  the  antecedent  be  worse  than  the  consequent  the 
mood  is  invalid. 

Rule  2. — If  the  antecedent  be  better  than  the  consequent  the 
mood  is  invalid. 

Definition. — Distributed  terms  are  those  modified  by  the  adjec- 
tives all  or  no,  i.  e.,  the  subject  of  a,  y  and  e  and  the  predicate  of 
a  and  €.  That  the  predicate  of  e  is  distributed  may  be  seen  at 
once  from  the  property  of  simple  convertibility  of  this  form. 

Rule  3. — If  a  term  which  is  distributed  in  the  consequent,  be 
undisturbed  in  the  antecedent,  the  mood  is  invalid. 

These  rules  are  both  necessary  and  sufficient  for  the  purpose 
which  they  effect,  as  will  appear  from  the  following  consideration : 
They  are  sufficient,  because  they  declare  all  the  moods  not  already 
found  to  be  valid,  to  be  invalid.  Each  one  is  necessary,  because 
we  can  point  to  at  least  one  invalid  mood,  which  falls  uniquely 
under  each  rule. 


CHAPTER   III 

§  12.  The  array  which  will  next  engage  our  attention  is  of  a 
somewhat  more  general  character  than  the  one  of  immediate 
inference,  and  we  may  begin  not  with  a  definition  of  it  in  abstract 
terms  but  with  an  examination  of  specific  examples. 

Let  us  consider  the  proposition,  a(ba)fi(cb)  Z  /3(ca),  A  single 
one  of  Euler's  diagrams  will  be  enough  to  furnish  a  representation 
of  the  two  factors  conjoined  in  the  antecedent. 


It  will  be  noticed  that  this  figure  verifies  not  only  the  antecedent 
but  the  consequent  as  well,  so  that  the  implication  would  appear 
to  be  a  valid  one.  It  may  be  observed  in  general,  that  whenever 
every  mode  of  representing  the  terms  as  related  in  the  antecedent 
is  also  a  mode  of  representing  the  terms  as  related  in  the  con- 
sequent, then  the  implication  is,  at  least  empirically  true. 

In  order  to  have  before  us  an  instance  in  which  this  double 
verification  breaks  down,  let  us  examine  the  proposition,  e(ab)  y  (be) 
Z  t(ca).  The  diagram  of  the  antecedent  as  a  whole  will  appear  as 
in  the  figure  below. 


But  here  there  exist  two  cases  in  which  the  consequent  is  not 
verified.  Accordingly,  if  a,  6  and  c  are  related  as  in  the  antecedent, 
it  does  not  follow  that  a  and  c  are  related  as  in  the  consequent  and 
the  empirical  untruth  of  the  implication  is  manifest. 

§  13.  The  type  of  proposition  which  our  two  examples  particu- 
larize, is  known  as  the  syllogism  and  its  general  expression  is 

19 


20 


Foundations  of  Formal  Logic 


x(a,  b)y(b,  c)  Z  z{c,  a).  Let  us  begin  our  study  of  this  form  of 
inference  by  determining  the  various  possible  ways  of  arranging 
the  terms.  These  varieties  of  figure  will  manifestly  be  not  more 
than  eight  in  number,  viz., 


ba 
cb 


ab 
cb 


ba 
be 


ab 
be 


ca 


ca 


ca 


ca 


ba 
cb 


ab 
cb 


ba 
be 


ab 
be 


ac 


ac 


ac 


ac 


The  two  forms,  x(a,  b)  and  y(b,  c),  conjoined  in  the  antecedent 
may  evidently  be  written  in  the  order  xy  or  in  the  order  yx,  for 
the  conjunctive  relation  of  logic  is,  as  we  say,  commutative.  Let 
it  now  be  agreed  as  a  matter  of  convention  always  to  write  first  the 
form  which  contains  the  predicate  of  the  consequent.  To  accord 
with  this  convention  the  second  set  of  four  term-orders  above 
will  have  to  be  rearranged  thus : 


cb 
ba 


cb 
ab 


be 
ba 


be 
ab 


ac 


ac 


ac 


ac 


Let  us  now  draw  a  horizontal  line  which  will  join  the  two  terms 
in  the  first  factor  of  the  antecedent  and  a  line  which  will  join  the 
term  b  above  with  the  term  b  below.  The  "figures ' '  so  constructed, 
when  isolated  from  the  letters  which  they  join,  would  then  appear 
thus: 


Perform  the  same  operation  again  for  the  set, 


ba 
cb 


ab 
cb 


ba 
be 


ab 
be 


ca 


ca 


ca 


ca 


Foundations  of  Formal  Logic 


21 


and  we  should  have,  allowing  the  letters  this  time  to  be  associated 
with  the  "figure,"  a  repetition  of  the  same  term-orders  but  differ- 
ently arranged,  viz., 


These,  accordingly,  contain  every  possible  distinction  of  term- 
order  and  will  be  constantly  referred  to  as  the  first,  second,  third 
and  fourth  figures  of  the  syllogism  respectively.  The  four  figures 
are  easily  remembered  as  combined  in  a  triangle  standing  on  its 
vertex. 


§  14.  Let  us  proceed  to  summarize  our  results  and  to  define 
certain  technical  expressions.  The  syllogism  is  a  form  of  impli- 
cation belonging  to  one  of  the  types, 

1.  x{ba)y{cb)  Z  z(ca) 

2.  x{ab)y{cb)  Z  z{ca) 

3.  x(ba)y(bc)  Z  z{ca) 

4.  x(ab)y(bc)  Z  z(ca) 

These  differences  are  known  as  the  four  figures  of  the  syllogism.' 
The  two  forms  conjoined  in  the  antecedent  are  called  the  premises 
and  the  consequent  is  called  the  conclusion.  The  predicate  of  the 
conclusion  is  called  the  major  term  and  points  out  the  major  premise, 
which  by  convention  is  written  first  in  the  antecedent.  The  subject 
of  the  conclusion  is  called  the  minor  term  and  points  out  the  minor 
premise.  The  term  which  is  common  to  the  premises  and  which 
does  not  appear  in  the  conclusion  is  called  the  middle  term. 

If  we  suppose  that  x,  y  and  z  mav  take  on  any  one  of  the 

1  Aristotle  recognized  only  the  first  three  figures  of  the  syllogism.  The 
discovery  of  the  fourth  is  due  to  Galen.  See  Prantl,  Geschichte  der  Logik  im 
Abendlande,  Leipzig,  1855,  Erster  Band,  p.  570. 


22  Foundations  of  Formal  Logic 

unprimed  letters,  a,  0,  7  and  c,  there  will  be  sixty-four  syllogistic 
variations  obtained  by  permuting  the  four  letters  three  at  a  time. 
Each  one  of  these  may  be  expressed  in  each  one  of  the  four  figures, 
so  that  two  hundred  and  fifty-six  cases  in  all  will  have  to  be  con- 
sidered. These  are  known  as  the  moods  of  the  array.  True  propo- 
sitions of  the  array  are  known  as  valid  moods  of  the  array.  The 
remainder  are  known  as  invalid  moods  of  the  array. 

In  constructing  the  array  of  the  syllogism,  it  will  prove  con- 
venient to  omit  the  implication  symbol,  Z ,  and  the  parts,  (b,  a), 
(c,  b)  and  (c,  a)  and  to  exhibit  each  mood  as  a  simple  combina- 
tion of  the  three  letters.  If  to  each  one  of  the  sixteen  permutations 
of  the  four  letters  taken  two  at  a  time,  each  one  of  the  four  letters 
be  added  in  succession,  the  array  under  each  figure  will  appear  thus : 

aaa  fiaa  70a  eaa 


0 

0 

0 

0 

7 

7 

7 

7 

e 

( 

€ 

e 

n/Ja 

m 

7/3a 

e/3a 

0 

0 

0 

0 

7 

7 

7 

7 

c 

c 

c 

6 

aya 

/37a 

77a 

67  a 

0 

0 

0 

0 

7 

7 

7 

7 

t 

6 

« 

e 

aea 

0ea 

7«a 

eea 

0 

0 

0 

0 

7 

7 

7 

7 

e 

£ 

c 

e 

Exercise. — Construct  the  array  of  the  syllogism  and  pick  out  by  the 
aid  of  Euler's  diagrams  the  valid  and  the  invalid  moods  under  each  figure. 

§  15.  As  was  done  in  the  case  of  immediate  inference,  we  shall 
now  formulate  rules  for  the  deduction  of  the  moods  of  the  syllo- 
gism. The  rules  and  postulates  below  will  yield  all  of  the  valid 
moods  under  each  figure. 

Definition. — Two  forms  are  said  to  be  alike  if  the  one  is  neither 
better  nor  worse  than  the  other.  The  members  of  all  other  pairs 
are  said  to  be  unlike. 

Rule  1 . — If  in  any  valid  mood  of  the  second  figure  a  like  minor 


Foundations  of  Formal  Logic         23 

premise  and  conclusion  be  each  made  singly  worse,  a  valid  mood 
will  result. 

Rule  2. — If  in  any  valid  mood  of  the  third  figure  a  like  major 
premise  and  conclusion  be  each  made  singly  worse,  a  valid  mood 
will  result. 

Rule  3. — If  in  any  valid  mood  the  subject  and  predicate  be 
interchanged  in  any  form  that  is  simply  convertible,  a  valid  mood 
will  result. 

Postulate  1. — a(ba)a(cb)  Z  a(ca)  is  a  valid  mood. 
Postulate  2. — 7(60)7(^:6)  Z  y(ca)  is  a  valid  mood. 
Postulate  3. — «(6a)7(c6)  Z  e(ca)  is  a  valid  mood. 
Theorems. — The  remaining  (26)  valid  moods. 

For  those  who  approach  the  study  of  the  syllogism  for  the  first  time 
it  may  be  well  to  point  out  the  effect  on  mood  and  figure  of  simple  con- 
version in  premise  or  conclusion. 

(a)  Simple  conversion  in  the  major  premise  changes  the  first  figure 
to  the  second  and  conversely,  the  third  figure  to  the  fourth  and  conversely. 

(b)  Simple  conversion  in  the  minor  premise  changes  the  first  figure  to 
the  third  and  conversely,  the  second  figure  to  the  fourth  and  conversely. 

(c)  Simple  conversion  in  the  conclusion  changes  the  first  figure  to  the 
fourth  and  conversely  and  leaves  the  second  and  third  figures  unchanged. 

It  must  of  course  not  escape  the  beginner's  notice  that  the  effect  of 
simple  conversion  in  the  conclusion  is  to  reverse  the  normal  order  of  the 
premises,  since  the  major  term  becomes  the  minor  term  and  the  minor 
term  becomes  the  major  term. 

Exercises 
In  the  exercises  below  the  mood  is  represented  as  a  simple  combination 
of  the  three  Greek  letters  and  the  figure  is  indicated  by  a  subscript. 

(1)  From  («7e)i  and  the  third  rule  alone  deduce  (eyeji,  {yt€)i  and 

(7«)«- 

(2)  From  (aoa)i,  (777)1  and  (077)1  and  the  third  rule  deduce  all  of 
the  remaining  valid  moods  by  the  aid  of  the  additional  principle:  If  in 
any  valid  mood  of  the  first  figure  a  like  major  premise  and  conclusion  be 
each  made  singly  worse,  a  valid  mood  will  result. 

(3)  Assuming  only  the  first  and  third  rules  deduce  the  remaining  valid 
moods  from  (aaa)i,  (777)1,  (707)1  and  («7€)i. 

The  rules  and  postulates  for  the  deduction  of  the  invalid  moods 
of  the  syllogism  are  listed  below.  In  enumerating  the  postulates, 
on  account  of  the  large  number  of  them,  the  abbreviation  of  the 
exercises  above  has  been  employed,  the  prime  over  the  bracket 
being  understood  as  standing  for  the  words,  "is  an  invalid  mood." 

Rule  1 . — If  in  any  invalid  mood  of  the  second  figure  a  like  minor 


24         Foundations  of  Formal  Logic 

premise  and  conclusion  be  each  made  singly  better,  an  invalid 
mood  will  result. 

Rule  2. — If  in  any  invalid  mood  of  the  third  figure  a  like  major 
premise  and  conclusion  be  each  made  singly  better,  an  invalid 
mood  will  result. 

Rule  3. — If  in  any  invalid  mood  of  the  fourth  figure  premises 
and  conclusion  which  are  all  alike,  be  each  made  singly  worse,  an 
invalid  mood  will  result. 

Rule  4. — If  in  any  invalid  mood  whose  premises  and  conclusion 
are  all  alike,  the  conclusion  be  made  better  or  worse  in  any  degree, 
an  invalid  mood  will  result. 

Rule  5. — If  in  any  invalid  mood  whose  premises  and  conclusion 
are  all  unlike,  either  premise  and  the  conclusion  be  interchanged, 
an  invalid  mood  will  result. 

Rule  6. — If  in  any  invalid  mood  the  subject  and  predicate  be 
interchanged  in  any  form  that  is  simply  convertible,  an  invalid 
mood  will  result. 

Postulates : 


(aafl'i 

(a/3e)'x 

(|S77)'3 

(777) '2 

(tfO'i 

(aery)'i 

(077/3 

GM'i 

(7C7)  '2 

(e77)'3 

(aat)\ 

(ac7)'i 

(7  cry) '2 

(ea7)'i 

to*)', 

(a/37)'i 

(/3a7)'i 

(7/37)'* 

(tfy)\ 

The  effect  on  mood  and  figure  of  interchanging  either  premise  and  the 
conclusion  (Rule  5)  may  be  conveniently  summarized  as  follows: 

(a)  Interchanging  major  premise  and  conclusion  changes  the  first 
figure  to  the  third  and  conversely  and  the  premises  remain  in  normal  order. 

(b)  Interchanging  minor  premise  and  conclusion  changes  the  first 
figure  to  the  second  and  conversely  and  the  premises  remain  in  normal 
order. 

(c)  Interchanging  major  premise  and  conclusion  changes  the  second 
figure  to  the  third  and  reverses  the  normal  order  of  the  premises. 

(d)  Interchanging  minor  premise  and  conclusion  changes  the  third 
figure  to  the  second  and  reverses  the  normal  order  of  the  premises. 

(e)  Interchanging  either  premise  and  conclusion  leaves  the  fourth 
figure  unchanged  and  reverses  the  normal  order  of  the  premises. 

Exercises 

(1)  From  (e/3e)'3  alone  deduce  seventy-eight  other  invalid  moods. 

(2)  From  (a/3e)'i  alone  deduce  twenty-three  other  invalid  moods. 

(3)  From  (a^y)'i  alone  deduce  eleven  other  invalid  moods. 

(4)  Deduce  the  invalid  moods  of  the  first  figure  which  have  a  7-minor 
premise. 


Foundations  of  Formal  Logic  25 

§  16.  As  in  the  case  of  immediate  inference  we  may  formulate 
rules  for  the  immediate  detection  of  the  invalid  moods  of  the  syllo- 
gism.    These  are  six  in  number. 

Rule  1. — A  mood  is  invalid  if  the  conclusion  be  unlike  two 
like  premises. 

Rule  2. — A  mood  is  invalid  if  the  conclusion  be  unlike  the  worse 
of  two  premises. 

Rule  3. — A  mood  is  invalid  if  each  premise  be  in  the  e-form. 

Rule  4. — A  mood  is  invalid  if  an  a-  and  a  7-premise  be  conjoined 
in  the  antecedent  and  the  middle  term  be  undistributed  in  the 
major  premise. 

Rule  5. — A  mood  is  invalid  if  the  middle  term  be  undistributed 
in  each  premise. 

Rule  6. — A  mood  is  invalid  if  a  term  which  is  distributed  in 
the  conclusion  be  undistributed  in  the  premise. 

The  rules  for  the  immediate  detection  of  the  invalid  moods  of 
the  syllogism  are  sufficient,  because  they  declare  all  the  moods 
not  already  found  to  be  valid  to  be  invalid.  They  are  all  necessary, 
because  we  can  point  to  at  least  one  example  which  falls  uniquely 
under  each  rule. 

Exercises 

(1)  Construct  the  array  of  the  syllogism  and  place  after  each  invalid 
mood  the  number  of  a  rule  that  declares  it  to  be  invalid. 

(2)  Show  that  it  follows  from  one  of  the  rules  alone  that  two 
/3-premises  do  not  imply  a  conclusion. 

(3)  Show  that  there  are  only  two  moods  which  illustrate  the  fourth 
rule  uniquely. 

(4)  Make  a  list  of  examples  which  fall  uniquely  under  each  one  of  the 
rules. 


CHAPTER  IV 

§  17.  The  type  of  implication  which  is  now  to  be  considered 
is  one  in  which  the  number  of  terms  is  greater  than  three  and,  as 
in  the  case  of  the  syllogism,  the  number  of  premises  one  less  than 
the  number  of  terms.  Accordingly,  it  will  be  more  convenient 
to  employ  in  place  of  the  class-symbols,  a,  b,  c,  etc.,  the  ordinal 
numbers,  1,2,3,  etc. 

The  sorites  is  an  implication  of  the  general  form : 

x{l,  2)y{2,  3)z{3,  4)..  .u(n  -  l,n)  Z  w(nl), 

in  which  n  is  greater  than  three  and  in  which  we  follow  the  con- 
vention of  writing  first  the  premise  which  contains  the  predicate 
of  the  conclusion. 

In  order  to  illustrate  the  manner  of  constructing  a  valid  mood 
of  the  sorites  from  a  chain  of  valid  syllogisms,  let  us  set  down  the 
following  definition  and  principles : 

Definition. — If  x  Z  y  is  a  valid  implication  then  x  is  said  to  be  a 
strengthened  form  of  y  and  y  is  said  to  be  a  weakened  form  of  x. 

Principle  i. — If  in  any  valid  mood  of  syllogism  or  of  sorites  a 
premise  be  strengthened  or  a  conclusion  be  weakened,  a  valid 
implication  will  result. 

Principle  ii. — If  in  any  valid  mood  of  syllogism  or  of  sorites  the 
same  factor  be  conjoined  to  both  antecedent  and  consequent  a 
valid  implication  will  result. 

Consider  the  chain  of  valid  syllogisms, 
y(21)y(32)  Z  y(31), 
y{31)y{43)  Z  y(41), 
y{41)y{54)  Z  y{51). 

Since  the  major  premise  of  the  last  member  of  the  chain  is  the 
same  as  the  conclusion  of  the  second  member,  it  may  be  strength- 
ened (by  principle  i)  to  y(31)y(43)  and  we  should  have: 
y{31)y{43)y{54)  Z  y{51) 

The  major  premise  of  this  result  is  in  turn  the  same  as  the 
conclusion  of  the  first  member  of  the  chain  and  may,  accordingly, 
be  strengthened  to  y(21)y(32).  The  valid  mood  of  the  sorites 
which  is  implied  by  the  chain  of  valid  syllogisms  is  therefore : 

y(21)y(32)y(43)y(54)  Z  y(51) 
26 


Foundations  of  Formal  Logic  27 

Suppose  that  we  were  to  begin  with  the  first  syllogism  of  the 
chain  and  were  to  conjoin  to  both  antecedent  and  consequent 
the  minor  premise  of  the  second  member  of  the  chain  (principle  ii) , 
i.  e., 

y(21)y(32)y(43)  Z  y{31)y{43) 

The  consequent  of  this  result  being  the  same  as  the  antecedent 
of  the  second  syllogism,  may  be  weakened  (principle  i),  to  y(41) 
and  we  should  have : 

y(21)y(32)y(43)  Z  y(41). 
Now  conjoin,  as  before,  to  antecedent  and  consequent  of  this 
mood  of  the  sorites  the  minor  premise  of  the  last  syllogism  of  the 
chain  (principle  ii),  i.  e., 

y(21)y(32)y(43)y(54)  Z  y{41)y{54) 
the  consequent  of  which  weakens  in  turn  (principle  i)  to  y(51)  so 
that  there  results 

y(21)y(32)y(43)y(54)  Z  y(51), 
the  same  sorites  as  the  one  obtained  by  the  process  which  was 
first  described. 

§  18.  In  general,  if  the  chain  of  syllogisms, 

*,(/,  2)x2(2,  3)  Z  x3(31), 
*3(j;)*4(J,  4)  Z  xs(41), 
xt(41)x«(4,  5)  Z  Xl(51), 


x2n-b(n  —  1  l)x2n.i(n  —  1,  n)  Z  x2n.3(nl). 
be  valid  throughout  in  each  of  its  members,  then 

xi(l,  2)x2(2,  3)xt(3,  4)  .  .  .  x2„.i(n  -  1,  n)  Z  x2n.3(nl) 
is  a  valid  mood  of  the  sorites.  Accordingly,  it  is  manifest  that  a 
certain  number  of  valid  moods  of  the  sorites  may  be  constructed 
from  chains  of  valid  syllogisms.  It  remains  to  be  proven  that 
the  only  valid  moods  of  the  sorites  that  exist  can  be  built  npjrom  chains 
of  valid  syllogisms  in  the  manner  described.  This  proof  depends 
upon  the  following  assumptions : 

Principle  i. — A  valid  mood  of  the  sorites  will  remain  valid  when 
as  many  terms  have  been  identified  as  we  desire. 

Principle  ii. — An  unprimed  a-premise,  whose  subject  and  predi- 
cate are  identical,  may  be  suppressed  as  a  unit  multiplier. 

Principle  iii. — A  valid  mood  of  the  sorites,  whose  premises  and 
conclusion  are  all  unprimed  forms  and  which  has  one  premise  of 


28         Foundations  of  Formal  Logic 

the  same  form  as  the  conclusion,  will  remain  valid  when  as  many 
other  premises  as  we  desire  are  put  in  the  a-form. 

Principle  iv. —  A  valid  mood  of  the  sorites,  whose  premises  and 
conclusion  are  all  unprimed  forms  and  none  of  whose  premises  has 
the  same  form  as  the  conclusion,  will  remain  valid  when  as  many 
premises  as  we  desire,  but  one,  are  put  in  the  a-form. 

Theorem  i. — There  exists  no  valid  mood  of  the  sorites  whose 
premises  and  conclusion  are  all  unprimed  forms  and  none  of  whose 
premises  has  the  same  form  as  the  conclusion. 

For  suppose  such  a  valid  mood  to  exist  and  put  all  the  premises 
after  the  first  two  in  the  a-form  (principle  iv).  Then  by  identi- 
fying terms  in  the  a-premises  so  obtained  (principle  i)  and  sup- 
pressing one  premise  after  another  (principle  ii) ,  we  should  in  the 
end  come  upon  an  invalid  syllogism.  Accordingly,  the  mood  of 
the  sorites  with  which  we  began,  must  be  invalid  as  well. 

In  the  general  solution  which  follows,  it  will  be  convenient  to 
take  the  conclusion  successively  in  each  one  of  its  four  possible 
forms. 

§  19.  Conclusion  in  the  a-form. 

At  least  one  of  the  premises  is  in  the  a-form  if  the  mood  of  the 
sorites  is  valid  (theorem  i).  If  one  of  the  remaining  premises 
x(s  —  1,  s)  be  not  in  the  a-form,  put  all  of  the  remaining  premises 
except  x  in  the  a-form,  if  they  be  not  already  in  that  form  (prin- 
ciple iii) .  Then  by  identifying  terms  (principle  i)  the  mood  of  the 
sorites  will  reduce  (principle  ii)  to  an  invalid  syllogism, 
x(s  -  1,  s)a(s,  s  +  1)  Z  a(s  +  1  s  -  1). 
or  a(s  —  2,  s  —  l)x(s  —  1,  s)  Z  a(s  s  —  2). 

Consequently,  all  of  the  premises  are  in  the  a-form,  if  the  mood 
of  the  sorites  is  valid  and  our  implication  must  be, 

a{l,  2)a(2,  3)  ...  a(n  -  1,  n)  Z  a(nl), 
which  can  be  constructed  from  the  chain  of  valid  syllogisms, 
a(l,  2)a(2,  3)  Z  a(31), 
a(31)a(3,  4)     Z  a{41), 


a(n  —  /  l)a{n  —  1,  n)  Z  a(n  1). 

§  20.  Conclusion  in  the  fi-form. 
At  least  one  of  the  premises  is  in  the  /S-form,  if  the  mood  is  valid 
(theorem  i),  and  all  of  the  other  premises  are  in  the  a-form.     For, 


Foundations  of  Formal  Logic  29 

suppose  one  of  the  other  premises  x(s  —  l,s)  were  not  in  the 
a-form.  Put  all  of  the  other  premises  except  x  and  the  /3-premise 
in  the  a-form  (principle  iii).  Then,  by  identifying  terms  (prin- 
ciple i),  the  mood  of  the  sorites  will  be  reducible  to  an  invalid 
syllogism  (principle  ii)  of  the  form, 

0(5  -  1,  s  -  2)x{s  -  l,s)  Z  |8(5  s  -  2), 
or  x(s  -  1,  s)0(s,  s  +  1)  /L  0(s  +  1  s  -  1). 
Consequently,  the  sorites  must  be 
a(l,  2)  ...  a(s,  s  -  l)0(s,  s  +  1)  .  .  .  a(n  -  1 ,  n)  Z  fi(n  1), 
and  this  may  be  built  up  out  of  the  chain  of  valid  syllogisms, 
a(l,  2)a(2,  3)   Z  a(31), 
a(31)a(3,4)     Z  a(41), 


a(s  -  1  l)a(s  -  l,s)  Z  a(sl), 
a(sl)P(s,s  +  1)  Z  /S(s  +  li), 
0(s  +  1  l)a(s  +  l,s  +  2)  Z  0(s  +  2  1), 


/3(w  -  1  l)a(n  -  1,  n)  Z  /3(wi). 

§  21.  Conclusion  in  the  y-form. 
At  least  one  of  the  premises  must  be  in  the  7-form,  if  the  mood 
of  the  sorites  is  to  be  valid  (theorem  i).  Moreover,  each  7-premise 
that  occurs,  must  present  its  terms  in  the  order  (s  s  —  l),i.e.,  with 
the  larger  ordinal  number  coming  first.  For  suppose  that 
y(s  —  Is)  should  appear  as  one  of  the  premises.  By  putting 
each  one  of  the  remaining  premises  in  the  a-form  (principle  iii) 
and  by  identifying  terms  (principle  i),  the  sorites  will  reduce  to 
an  invalid  syllogism  (principle  ii), 

7(s  -  1  s)a(s,  s  +  1)  Z  y(s  +  1  s  -  1), 
or  a(s  —  l,s  -  2)y(s  —  Is)  Z  y(s  s  —  2). 
Just  as  in  the  cases  already  considered  it  can  be  shown  that  no 
/3-  or  €-premise  can  occur.     One  valid  mood  of  the  sorites  may 
consequently  be 

y(21)y(32)  ...  y(n  n  -  1)  Z  y(n  1), 
which   can,   in   fact,   be   constructed   from   the   chain   of   valid 
syllogisms, 

y{21)y(32)  Z  y(31), 
y(31)y(43)  Z  y(41), 


y{n  -  1  l)y(n  n  -  1)  Z  y{n  1). 


30         Foundations  of  Formal  Logic 

It  will  now  be  manifest  that  all  the  other  forms  of  valid  sorites 
with  a  7-conclusion  are  to  be  gotten  from  the  above  type  by  trans- 
forming one  or  more  of  the  premises  into  the  a-form  in  every 
possible  way  under  the  restrictions  of  theorem  i  and  it  will  be 
easy  in  each  case  to  construct  the  chain  of  generating  syllogisms. 

§  22.  Conclusion  in  the  e-form. 

At  least  one  of  the  premises  is  in  the  e-form  (theorem  i)  and 
there  is  not  more  than  one  e-premise.  For,  if  there  were  two  or 
more  e-premises,  we  might  put  all  but  two  of  the  e-premises  in 
the  a-form  (principle  iii).  Then,  by  identifying  terms  (principle  i) 
we  should  come  upon  an  invalid  syllogism  (principle  ii), 
e(s  -  1,  s)e(s,  S  +  l)  Z  e(s  +  /  s  -  1). 

There  can  be  present  no  /3-premise.  For  suppose  one  or  more 
such  premises  to  be  present.  By  putting  all  of  the  premises  except 
one  /3-premise  and  the  e-premise  in  the  a-form  (principle  iii),  we 
should  by  identifying  terms  (principle  i)  come  upon  an  invalid 
syllogism  (principle  ii), 

/3(s  -  /,  s)e(s,  s  +  1)  Z  e(s  +  1  s  -  1), 
or  e(s  -  2,  s  -  i)/3(s  -  l,s)  Z  e(s  5  -  2). 

Any  7-premise  coming  after  the  e-premise  must  present  its  terms 
in  the  order  (s  s  —  1).  For  suppose  y(s,  s  —  1)  coming  after 
the  epremise,  to  present  the  term-order  (s  —  1  s).  Putting  all 
the  premises  except  y(s  —  1  s)  and  the  e-premise  in  the  a-form 
(principle  iii)  we  should  by  identifying  terms  (principle  i)  come 
upon  an  invalid  syllogism  (principle  ii), 

e(s  -  l,s  -  2)y(s  -Is)  Z  e(s  s  -  2). 

Any  7-premise  coming  before  the  e-premise  must  present  its 
terms  in  the  order  (s  —  1  s).  For  suppose  7(5,  s  —  1),  coming 
before  the  e-premise  to  present  the  term-order  (s  s  —  1).  Putting 
all  of  the  premises  except  7(5  s  —  1)  and  the  e-premise  in  the 
a-form  (principle  iii),  we  should  by  identifying  terms  (principle  i) 
come  upon  an  invalid  syllogism  (principle  ii), 

7(5  5  -  l)e(s,  s  +  1)  Z  e(s  -f  1  s  -  1). 

Consequently,  one  valid  mood  of  the  sorites  may  be 
y(12)    ...    7(5  -  1  s)t{s,  s  +  l)y(s  +  2  s  +  1)    ...    y(n  n  -  1) 
Z  e(n/) 

which   can,   in   fact,   be   constructed   from   the   chain   of  valid 
syllogisms, 


Foundations  of  Formal  Logic  31 

y(12)y(23)  Z  y(13), 
y(13)y(34)  Z  y{14), 


y(l  s  -  J)y(s  -Is)  Z  y(l  s), 
y(l  s)t(s,  s  +  1)  Ze(s  +  1  1), 
e(s  +  l  l)y(s  +  2s  +  l)  £.  c(s  +  2  /), 

§(*  -  1  J)y(n  n  -  1)  Z  c(«  /). 

All  the  other  forms  of  valid  sorites  with  an  ^-conclusion  are 
clearly  to  be  gotten  from  the  type  that  has  just  been  established 
by  putting  one  or  more  of  the  7-premises  into  the  a-form,  and  it 
will  be  easy  in  each  case  to  construct  the  corresponding  chain  of 
generating  syllogisms. 

There  exist  consequently  no  valid  moods  of  the  sorites,  whose 
premises  and  conclusion  are  all  unprimed  forms,  which  cannot  be 
built  up  out  of  chains  of  valid  syllogisms. 

When  the  general  solution  of  the  sorites  has  been  effected  by 
means  of  the  principles  of  exercise  (5)  below,  i.  e.,  when  all  valid 
implications  of  the  form, 

x(l,  2)y{2,  3) z{n  -  l,n)  Z  w{nl), 

have  been  determined,  we  shall  have  resolved  at  the  same  time 
a  type  of  inference  composed  of  the  product  of  n  premises,  con- 
taining a  cycle  of  n  terms  and  implying  zero,  viz., 
x{l,  2)y{2,  3)  ....  z(n,  1)  Z  0. 
That  the  solution  of  this  last  type  is  exactly  equivalent  to  the  one 
given  follows  from  the  principles : 

(*  Z  y)  Z  (%/  Z  0), 
\xy'  Z  0)  Z  (*  Z  y). 

Exercises 

(1)  Construct  a  valid  mood  of  the  sorites  from  the  following  chain  of 
valid  syllogisms, 

a(21)y(32)  Z  y{31), 
y(31)a(43)  Z  y(41), 
y{41)y{54)  Z  y(51). 

(2)  Employing  the  principles  of  this  chapter,  reduce  the  sorites, 

a(21)y(32)a(43)y(54)  Z  7(5/), 
successively  to  each  one  of  the  three  valid  syllogisms  of  exercise  (1). 

(3)  Employing  the  principles  of  this  chapter,  establish  the  invalidity 
of  the  sorites, 

y(21)y(32)y(34)y(54)  Z  y(51). 


32         Foundations  of  Formal  Logic 

(4)  From  what  chain  of  valid  syllogisms  can  the  sorites, 

y(12)y(23U34)y(54)  Z  6(5/), 
be  built  up? 

(5)  Complete  the  general  solution  of  the  sorites  begun  in  this  chapter, 
taking  for  granted  the  following  principles: 

Principle  v. — A  valid  mood  of  the  sorites,  whose  premises  are  all  unprimed 
forms  and  whose  conclusion  is  a  primed  form  and  all  of  whose  premises 
and  conclusion  are  of  the  same  form,  will  remain  valid,  when  as  many 
premises  as  we  desire,  but  one,  are  put  in  the  a-form. 

Principle  vi. — A  valid  mood  of  the  sorites,  whose  premises  are  all 
unprimed  forms  and  whose  conclusion  is  a  primed  form  and  one  of  whose 
premises  is  a  form  different  from  the  conclusion,  will  remain  valid,  when 
as  many  other  premises  as  we  desire,  are  put  in  the  a-form. 


CHAPTER  V 

§  23.  In  the  chapters  which  follow,  the  solutions  already  given 
will  be  generalized,  new  results  will  appear  and  the  whole  will  be 
more  completely  expressed  in  the  symbolical  language  of  an 
algebra.1 

A  principle,  which  is  altogether  fundamental  and  which  will  be 
taken  for  granted  at  each  step  of  our  progress,  is  this :  //  a  propo- 
sition is  true  in  general,  it  is  because  it  remains  true  for  all  specific 
meanings  of  the  terms  that  enter  into  it,  although  an  untrue  propo- 
sition does  not  always  remain  untrue  in  the  same  circumstances. 
Thus  y(ab)  Z  7 (6a)  is  untrue  in  general,  but  it  becomes  true  for 
the  case,  in  which  a  and  b  are  identical,  viz.,  y(aa)  Z  y(aa). 
Accordingly,  when  we  write  {x(a,  b)  Z  y(a,b)\',  we  only  assert 
that  there  is  at  least  one  value  of  a  and  one  value  of  b  which  will 
render  x{a,  b)  Z  y(a,  b)  a  false  proposition.  If  it  has  been  estab- 
lished that  x{aa)  Z  y(aa)  is  untrue,  then  we  may  at  once  infer 
that  the  more  general  implication,  x(a,  b)  Z  y{a,  b)  is  untrue  as 
well.  In  order  to  establish  the  untruth  of  a  given  proposition,  it  will 
be  enough  to  point  to  a  special  instance  of  its  being  untrue. 

"The  Implicative  Function  is  a  propositional  function  with  two  argu- 
ments p  and  q,  and  is  the  proposition  that  either  not-/)  or  q  is  true,  that 
is,  it  is  the  proposition  p'  +  q.  Thus  if  p  is  true,  p'  is  false,  and  accord- 
ingly the  only  alternative  left  by  the  proposition  p'  +  q  is  that  q  is  true. 
In  this  sense  the  proposition  p'  +  q  will  be  quoted  as  stating  that  p  implies 
q."  (Whitehead  and  Russell,  Principia  Mathemalica,  Cambridge  Uni- 
versity Press,  1910,  vol.  1,  p.  7.)  This  limitation  we  do  not  allow,  for 
it  will  be  found  to  run  counter  to  the  whole  meaning  of  the  system  we  are 
building  up.  In  the  citation  above  we  have  replaced  the  authors'  nota- 
tion by  our  own. 

§  24.  In  presenting  the  materials  of  our  subject-matter  we 
shall  have  to  deal  with  two  types  of  proposition.     The  truth  of 

(x  Zy)(y  Iz)  Z  (*  Z  z) 
is  independent  of  x,  y  and  z,  no  matter  for  what  propositions 

1  Symbolic  logic  was  first  developed  by  Boole  in  a  work  entitled  An  Investi- 
gation of  the  Laws  of  Thought,  1854.  Leibnitz  (1646-1716)  in  his  youth  had 
suggested  the  plan  of  such  a  calculus  but  he  never  executed  it.  Toward  the 
end  of  his  life  he  remarked  that  in  such  a  universal  language  the  characters 
or  symbols  would  direct  the  operations  of  thought  and  all  error  would  be 
reduced  to  error  of  computation. 

33 


34         Foundations  of  Formal  Logic 

x,  y  and  z  may  stand.     Such  a  general  truth  will  be  termed  a 
principle.    The  truth  of 

y(ba)y(cb)  Z  y(ca) 
is  independent  of  a,  b  and  c,  no  matter  for  what  classes,  a,  b  and  c 
may  stand.     If  such  a  general  truth  has  to  be  taken  for  granted, 
it  will  be  termed  a  postulate. 

Principles  are,  accordingly,  independent  of  forms;  postulates 
are  independent  of  terms. 

§  25.  We  begin  by  setting  down  five  postulates,  the  truth  of 
which  should  be  verified  again  empirically  by  the  familiar  device 
of  Euler's  circular  diagrams. 

(i)       a(ba)a(bc)  Z  a(ca), 

(ii)      a(6a)/3(6c)  Z  0(ca), 

(iii)     a(ba)y(cb)  Z  y(ca), 

(iv)     a(6a)c(6c)    Z  t{ca), 

(v)      y{ba)a{bc)  Z  y{ca), 
and  we  shall  add  to  these 

(vi)  a'(aa)  Z  a(ao). 
This  last  assumption  illustrates  an  extension  of  the  common 
meaning  of  implication  and  is  forced  upon  us,  if  we  are  to  allow 
a(aa)  to  stand  for  a  true  proposition.  The  uses,  to  which  such  an 
extension  of  meaning  may  be  put,  will  become  clear  in  the  sequel. 
It  will  be  enough  to  state  that  a  proposition,  which  is  true  for  all 
meanings  of  the  terms,  will  be  implied  by  the  proposition  whose 
symbol  is  i,  and  behaves  like  a  unit  multiplier  in  this  algebra. 

Only  a  small  number  of  the  principles,  which  we  shall  introduce 
as  necessity  requires,  are  independent,  but  it  will  not  concern  our 
purpose  to  point  out  the  manner  of  their  inter-connection. 
From  the  principle, 

(%'  Z  *)  Z(y  Z  *), 
we  obtain,  by  (vi),  the  theorem, 

i  Z  a(aa). 
By  (i),  for  a  =  b,  a(aa)a(ac)  Z  a(ca),  and 

[a(aa)a(ac)  Z  a(ca)}   {i  Z  a{aa)\  Z  \a{ac)  Z  a(ca)\, 
by  {xy  Z  z)  {w  Z  x)  Z  (wy  Z  z), 

omitting  the  unit  multiplier,  i,  as  it  is  the  custom  to  do.    Similarly, 
we  obtain 

/3(ac)  Z  /3(ca),  from  (ii), 

y(ca)  Z  y(ca),  from  (iii), 

«(oc)  Z  e(ca),  from  (iv). 


Foundations  of  Formal  Logic         35 

Again, 

U(c6)  Z  a(ba))  [a(ba)  Z  a(ab))   Z  \a(ab)  Z  a(ab)\, 
by  (x  Z  y)  (y  Z  s)  Z  (%  Z  «). 

Accordingly, 

a(a6)  Z  0(06), 

0(a&)  Z  0(a&), 

7(a6)  Z  7(06). 

«(a&)  Z  «(o6). 

A  complete  induction  of  the  members  of  this  set  and  an  applica- 
tion of  the  principle, 

(x  Z  y)  Z  (yf  Z  *'), 

yields  the  general  result, 

I.     k'(ab)  Z  k'(a6), 
k(a&)  being  understood  as  representing  any  one  of  the  unprimed 
letters,  a,  0,  7,  €. 

§  26.  Each  one  of  the  propositions  so  far  derived,  being  true,  is 
implied  by  the  unit  multiplier,  i.  The  contradictory  (as  it  is  called) 
of  i  will  be  any  proposition,  which  is  untrue  for  all  meanings  of  the 
terms  that  enter  into  it.  It  will  be  represented  by  the  symbol,  0, 
and  will  be  denned  by 

0  Z  i,         (*  Z  0)', 
wherein  it  will  be  seen  that  i  stands  for  0'.    For  a  verbal  interpreta- 
tion of  o  and  i  we  may  read : 

o  =  No  proposition  is  true, 

i  =  One  proposition  is  true. 

§  27.  The  utility  of  the  concept  of  zero,  0  (the  Mw//-proposition) , 
and  of  one,  i  (the  on^-proposition),  will  be  illustrated  in  part  by 
the  derivations,  which  follow.     We  have 

{a(ab)  Z  a(ab)\   Z  U(a&)a'(a&)  Z  o\, 
by  {%  Z  y)  Z  (x  y'  Z  o); 

{a(ab)a'(ab)  Z  0}   Z  (a(a&)  Z  a"(a&)}, 
by  (*v  Z  0)  Z  (x  Z  /). 
Thus  we  should  obtain 

a(ab)  Z  a"(a&), 
P(ab)  Z  /3"(a&), 
7(06)  Z  y"(ab), 
e(ab)    Z   «"(a&). 


36         Foundations  of  Formal  Logic 

\a(ab)  Z  a(ab)\   Z  \a'(ab)  Z  a'(ab)\, 
by  (x  Z  y)  Z  (/  Z  %') ; 

{a'(ab)  Z  a'(ab)}   Z   U"(a&)  Z  a(a&)}, 
by  (%'  Z  y)  Z  (/  Z  %). 
Accordingly, 

a"{ab)  Z  a(a6), 
/3"(a&)  Z  0(a&), 
7"(a6)  Z  T(a&), 
e"(ab)    Z   e(a&), 

and  a  general  result  will  be  obvious,  viz., 

II.     k(ab)  Z  k"(a&),        k"(ab)  Z  k(a&), 
wherein  the  same  restriction  is  imposed  upon  k(a6)  as  before. 

§  28.  The  principle  that  the  truth  of  any  one  of  the  four  cate- 
gorical forms  implies  the  falsity  of  each  one  of  the  others,  a  general- 
ization which  will  now  be  established,  is  characteristic  of  the  logic 
which  we  are  constructing.  We  shall  begin  by  setting  down  the 
three  characteristic  postulates, 

(vii)      /3(aa)  Z  P'(aa), 

(viii)     7(aa)  Z  y'(aa), 

(ix)       t{aa)    Z   c'(aa). 

Then,  by  the  principle, 

0  Z  %')  Z  (x  Z  y), 

we  may  establish  at  once 

(3(aa)  Z  o, 
y(aa)  Z  <?, 
e(aa)    Z  o. 
Postulate    (ii)    above    yields,    for    a  =  c,    a(ba)(l{ba)  Z  /3(aa); 
{a(6a)^(6a)  Z  0(aa)}  (/3(aa)  Z  o\  Z  {a(6a)/3(6a)  Z  <?}, 
by  (x  Zy){y  Z  z)  Z  (x  Z  z); 

{a(6a)/3(6a)  Z  o)  Z  \a(ba)  Z  /3'(6a)}, 
by  {xy  Z  6)  Z  (x  Z  y'). 

Similarly,  by  (iii),  a(ba)  Z  y'(ab); 

{a(ab)  Z  a(ba)\   \a(Jba)  Z  y'(ab)}   Z  {a(ab)  Z  y'(ab)\ 
by  (x  Z  y)  {y  Z  z)  Z  (x  Z  z) ; 
and,  by  (iv),  we  obtain  a(ab)  Z  e'(ab). 
If  now  we  postulate, 

(x)     y(ba)y(cb)    Z  y(ca), 
(xi)     e(ba)y(cb)    Z   e(ca), 


Foundations  of  Formal  Logic         37 

there  result  as  before  t(ab)  Z  y'(ab)  and  y(ab)  Z  y'(ba),  and  if 

(xii)      /8(o6)  Z  y'(ab), 

(xiii)     0(a&)  Z   e'(a&), 
all  of  the  remaining  implications  of  this  form,  viz., 

t(ab)  Z  a'(a&)  7(06)  Z  a'(ab) 

e(ab)  Z  j8'(a6)  7(a6)  Z  0'(a&) 

e(a&)  Z  7'(a6)  0(a6)  Z  a'(a6) 

are  obtained  at  once  from  those  that  have  just  been  established  by 

(*  Z  </)  Z.{y  Z  *')• 

If,  now,  k(a&)  and  w(a6)  can  not  represent  the  same  categorical 

form,  y(ab)  and  7(60)  being  considered  distinct,  and  if  further 

k(ab)  and  w(ab)  can  stand  only  for  the  unprimed  letters,  a,  /3,  7 

and  e,  then, 

III.     k(a&)  Z  w'(a&). 
§  29.  We  shall  now  establish  the  untruth  of  certain  forms  of 
implication,  making  them  ultimately  depend  upon  the  invalidity 
of  *  Z  0,  whose  untruth  is  set  down  as  a  matter  of  definition.2 
Suppose  a(aa)  Z  a'(aa)  were  true. 

[i  Z  a(aa)}   {a(aa)  Z  a'(aa)}   Z   [i  Z  a'(aa\, 
by  (x  Z  y)  (y  Z  z)  Z  (x  Z  z) ; 

(•  Z  a'(oa)}   (a'(aa)  Z  oj   Z  {:'  Z  o\ , 
by  the  same  principle. 
But  i  Z  0  is  untrue. 
.'.  a(aa)  Z  a'(aa)  is  untrue  as  well. 
Again, 

U(aa)  Z  a'faa)}'  {j8(ao)  Z  a'(aa)\   Z  (a(aa)  Z  /8(aa)}', 
by  (x  Z  2)'  (v  Z  2)  Z  (%  Z  y)'\ 

{a(aa)  Z  /3'(aa)}  (a(aa)  Z  /8(aa}'  Z  {/3'(aa)  Z  /3(aa}', 
by  (x  Z  y)  («Z  2)'  Z(jZ  2)'. 
Accordingly,  we  have 

\a(aa)  Z  o'(ao)|', 
{/3r(aa)  Z  /3(aa)|'. 
|7'(aa)  Z  7(aa)}'. 
|€'(aa)  Z  «(aa)}'f 
and,  since  the  untruth  of  any  proposition  is  implied,  whenever  we 
can  point  to  a  special  instance  of  its  being  untrue,  it  follows  that 

2  Some  of  the  possibilities  of  the  method  of  reductio  ad  absurdum  in  logic 
were  pointed  out  and  elaborated  by  the  famous  Italian  geometer  Saccheri 
(Logica  d-emonstrativa,  1697). 


121 


38  Foundations  of  Formal  Logic 

{a(ab)  Z  a'(ab)Y, 
lfi'(ab)  Z  0(db)V, 
Wiab)  Z  y(ab))', 
Wiab)  Z  *(ab)}'. 
The  first  member  of  the  set 

Wiab)  Z  a(ab)}'t 
\0(ab)  Z  P'(ab)}', 
{y(ab)  Zy'(ab)}', 
U{ab)  Z  e'(a&).}', 
may  be  gotten  from  one  of  the  other  three,  thus, 

\y(ab)  Z  y\ab))'  [a(ab)  Z  y'(db))  Z  \y(ab)  Z  a(ab))', 
by  {x  Z  «)'  (yZ2)Z(«Z  ?)'; 

\y(ab)  Z  o'(a6)}  {7(a6)  Z  a(ab)}'  Z  (a'(a&)  Z  a(a6)}', 
by  (x  Z  y){x  Z  z)'  Z  (y  Z  z)'. 

For  the  reduction  of  the  last  three  see  exercise  (10)  at  the  end  of 
this  chapter. 

As  a  result  of  a  complete  induction  of  the  members  of  these  sets 
and  upon  application  of  (x  Z  y)'  Z  (y'  Z  x')f,  it  follows  that 

IV.     \k(ab)    Z  k'(a&)}\  \k'(ab)  Z   k(a&)}\ 

\k'(ab)  Z  k"(ab)}',  {k"(ab)  Z   k'(a6)}'. 

§  30.  If  a'(ab)  Z  /3(a6)  were  a  true  implication,  we  should  have: 
{y(ab)  Z  a'(ab)\  [a'(ab)  Z  P(ab)}  Z  {y(ab)  Z  /J(o6)J, 
by  (x  Z  y)  (y  Z  z)  Z  (x  Z  z); 

{y(ab)  Z  fi(ab))  (j8(o&)  Z  y'(ab)}  Z  \y(ab)  Z  y'(ab)) 
by  the  same  principle. 

.\a'(a&)  Z  0(ab)  is  untrue. 

Applying  the  same  method  of  reduction  there  will  result : 

Wiab)  Z  p(ab))',  \0\ab)  Z  y(ab)Y, 

Wiab)  Zy(ab)Y,  \P'(ab)  Z   e(ab)Y, 

Wiab)  Z  t(ab)Y,  Wiab)  Z  e(o6)}'. 

Wiab)  Zy(ba)}', 

and  upon  application  of 

(%'  Z  y)'  Z  (/  Z  x)', 

Wiab)  Z  a(ab)Y,  Wiab)  Z  a(ab)Y, 

Wiab)  Z  P(ab)Y,  Wiab)  Z  P(ab)Y, 

U\ab)  Zy(ab)Y,  Wiab)  Z  o(o6)}'. 

We  are  now  prepared  to  lay  down  the  final  generalizations  which 


Foundations  of  Formal  Logic         39 

are  given  below.      From  the  propositions  that  have  just  been 
enumerated  there  will  follow 

V.     |w'(a&)  Z  k(ab)\'; 
from  III  and  V,  by 

(*  Z  y)   Z  (/  Z  x') 

(x  Z  y)'  Z  (/  Z  x')' 

VI.  k"(a6)  Z  w'(a&),  |w'(a&)  Z  k"(a&)}'; 
from  III  and  IV,  by 

(x  Z  z)'{y  Zz)  Z  (%  Z  y)', 
(*  Z  y)'  Z  (/  Z  *')', 

VII.  ik(ab)  Z  w"(a&)}',  (w^afc)  Z  k(ab)\'. 

§31.  In  order  to  classify  the  categorical  forms  under  the  heads, 
contradictories,  contraries,  subcontraries ,  and  subalter>is,  let  us 
consider  what  special  meanings  of  x{ab)  and  y(ab)  render  true 
or  untrue, 

(1)  x(ab)    Z  y'(ab), 

(2)  y'{ab)  Z   x(ab). 

If  x(ab)  and  y(a6)  satisfy  (1)  and  (2)  together,  x(ab)  is  said  to 
be  contradictory  to  y{ab).  By  I,  k'(ab)  is  contradictory  to  k(ab) 
and,  by  II,  k(ab)  is  contradictory  to  k'(ab). 

If  #(a6)  and  y(ab)  satisfy  (1)  alone,  x(ab)  is  said  to  be  contrary 
to  y(ab).     By  III  and  V,  k(ab)  is  contrary  to  w(a6). 

If  x(a&)  and  y(a£)  satisfy  (2)  alone  x(ab)  is  said  to  be  subcontrary 
to  y(ab).     By  VI,  k'(at)  is  subcontrary  to  w'(a6). 

If  %(a&)  and  y(ab)  satisfy  neither  (1)  nor  (2),  x{ab)  is  said  to  be 
subaltern  to  y(ab).  By  VII,  k(ab)  is  subaltern  to  w'(ab),  and,  by 
IV,  k(ab)  and  k'(a&)  are  each  the  subalterns  of  themselves. 

Exercises 
1.  The  meaning  of  logical  equality  is  given  by 

(x  Z  y)(yZ   x)  Z  (x  =  y), 

(x=y)Z  (*  Zy)(yZ.). 
If  k(ab)  =  k(ab)k(ab)  and  k(ab)  Z  w'(ai),  show  that 

a(ab)  Z  /3'(a6)7,(o6)e'(a6)7/(M, 
/3(a6)  Z  a'(ab)y'(ab)t'(ab)y'(ba), 
y(ab)  Z  a'(a6)i8'(a6)€'(a6)7'(ta), 
«(a&)  Z  o'(a6)/3'(a6)7'(aft)7'(^), 
by  the  aid  of 

(x  Z  y)(y  Z  z)  Z  (*  Z  «), 

(x  Z  y)  Z  (zx  Z  zy). 


40  Foundations  of  Formal  Logic 

2.  If 

0'(ab)y'(ab)e'(ab)y'(ba)  Z  a(ab), 
a'(ab)y'(ab)e'(ab)y'(ba)  Z  &(ab), 
a'(ab)p'(ab)<:'(ab)y'(ba)  Z  y(ab), 
a'(ab)(3'(ab)y'(ab)y'(ba)  Z  e(ab), 
establish 

a'(ab)  =  $(ab)  +  y{ab)  +  e(ab)  +  y(ba), 
P'{ab)  =  a(ab)  4-  y(ab)  4-  e(ab)  -f  y(ba), 
y'(ab)  =  a(ab)  4-  fi(ab)  +  e(ab)  +  y(ba), 
t'{ab)  =  a(ab)  4-  P(ab)  +  7(a&)  +  y(ba), 
assuming  that  the  contradictory  of  a  product  is  the  sum  of  the  contra- 
dictories of  the  separate  factors  and  assuming  the  right  to  substitute 
k(ab)  directly  for  k"(ab). 

3.  Assuming  x(ab)  =  x(ab)x(ab),  x{ab)  =  x(ab)  +  x{ab),  show  that 

a'(ab)P'(ab)  =  y(ab)  4-  e(ab)  4-  7(60),  etc.,  etc., 
a(ab)  4-  P(ab)  =  y'{ab)t'(ab)y'(ba),  etc.,  etc. 

4.  Establish  the  general  results, 

k(ab)  =  k(ab)w'(ab),        k'(ab)  =  k'(ab)  4-  w(ab), 
k(ab)w(ab)  =  0. 

5.  From  the  principle,  (x  Z  z)'(y  Z  2)  Z  (x  Z  y)',  and  the  postulate, 
(a(oo)  Z  a'(aa)}',  derive  {a(<za)  Z  /3(ao)}',  { a(ac)  Z  7(aa)}',  {a(ao) 
Z  e(aa)}\ 

6.  By  the  aid  of  the  principles, 

(x  Z  y)(y  Z  2)  Z  (x  Z  2),         (jc  Z  *')  Z(iZ  y), 
from  the  postulate,  a'(aa)  Z  a(ao),  and  results  already  established,  (viz., 
Ill),  show  that  all  propositions  of  the  form,  x(aa)  Z  y(aa),  except  the 
three  cases  in  the  last  example,  are  true  implications,  x(aa)  and  y(aa) 
representing  only  the  unprimed  letters. 

7  Show  by  the  method  of  the  last  example  that  a(aa)  Z  a'(aa)  is  the 
only  untrue  implication  of  the  form  x{aa)  Z  y'(aa). 

8.  Derive  seven  true  implications  of  the  form,  x'(aa)  Z  y(aa),  and  nine 
untrue  implications  of  the  same  form. 

9.  Establish  the  untruth  of 

k(a,  b)  Z  w(a,  b),        k'(a,  b)  Z  w'(a,  b). 

10.  Establish  the  untruth  of  P(ab)  Z  0'(a&)  and  e(ab)  Z  e'(ab),  by 
making  them  depend  upon  the  untruth  of  (3(ba)(l(cb)  Z  /S'(ca)  and 
t(ba)t(cb)  Z  e'(co)  respectively  (see  the  postulates  of  the  next  chapter). 
Thus, 

[PibaMcb)  ^  &(.ca)\'  Z  {j8(fta)j8(c6)/3(ca)  Z  0}', 
by  (*  Z  y')'  Z  (xy  Z  0)'; 
|/3(6a)/3(c6)/3(ca)  Z  o\'{P(cb)0(ca)o  Z  0}  Z. 

[p(ba)P(cb)P(ca)  Z  0(c&)/3(ca)o}', 
by  (xy  Z  2)'(w  Z  2)  Z  (xy  Zw)'; 

{j8(6fl)j3(c6)/3(ca)  Z  /3(c&)/3(ca)o}'  Z  |j8(6a)  Z  a}', 
by  (xz  Z  2y)'  Z  (x  Z  y)'; 

||8(6a)  Zo|'Z  {0(6a)  Z/3'(M}', 
by  (*  Z  0)'  Z(iZ  *')'. 


CHAPTER   VI 

§  32.  The  valid  moods  of  the  syllogism, 

x(a,  b)y(b,  c)  Z  z(ca), 
twenty-nine  in  all,  which  are  not  set  down  among  the  assumptions 
of  the  last  chapter,  may  be  derived  at  once  by  the  following 
principles : 

(xy  Z  z)  (w  Z  x)  Z  (wy  Z  z), 
(x  Z  y)  (y  Z  z)  Z  (x  Z  z), 
(xy  Z  z)  Z  (yx  Z  z). 

Thus,  from  postulate  (xi),  Chap.  V,  by  the  second  principle, 

U(ba)y(cb)  Z  e(ca)}   U(ca)  Z  e(ac)}   Z   {e(6ah(Vfc)  Z  «(ac) } , 
and,  since  the  term-order  in  the  conclusion  is  now  reversed,  so  that 
the  major  term  has  become  the  minor  term  and  the  minor  term  has 
become  the  major  term,  it  will  be  necessary  to  employ  the  third 
principle  to  restore  the  normal  order  of  the  premises.    Accordingly, 

U(ba)y(cb)  Z  e(ac)\   Z  \y(cb)  e(ba)  Z  t(ac)\, 
and  it  will  be  seen  that  the  term  order  in  this  result  is  that  of  the 
fourth  figure.     The  second  principle  (above)  thus  enables  us  to 
convert  simply  in  the  conclusion  and  the  effect  of  simple  conversion 
in  the  conclusion  is  to  change  the  first  figure  to  the  fourth. 

Similarly,  since  the  third  principle  enables  us  to  arrange  the 
premises  in  either  order,  the  first  principle  will  allow  us  to  convert 
simply  in  either  premise,  if  that  premise  be  not  in  the  7-form. 
Thus,  from  postulate  (i)  of  the  last  chapter 

\a(ba)a(bc)  Z  a(ca)\   Z   \a(bc)a(ba)  Z  a(ca)\, 
by  the  third  principle  (above) ; 

{a(bc)a(ba)  Z  a(ca)\   \a(cb)  Z  a(bc)\   Z   \a(cb)a(ba)  Z  a(ca)\, 
by  the  first  principle  (above) ; 

\a(cb)a(ba)  Z  a(ca)\   Z  [a(ba)a(cb)  Z  a(ca)\, 
by  the  third  principle  (above) ;  and  this  result  is  a  valid  mood  of 
the  first  figure.     However,  when  it  is  desired  to  convert  simply  in 
the  minor  premise,  it  will  be  more  convenient  to  employ  at  once 
the  principle, 

(xy  Z  z)  (w  Z  y)  Z  (xvu  Z  z), 
and  avoid  two  of  the  three  steps,  that  would  otherwise  be  necessary. 

41 


42  Foundations  of  Formal  Logic 

Exercise 

From  postulates  (i-v)  and  postulate  (xi)  of  the  last  chapter  derive 

twenty-two  valid  moods  of  the  syllogism  by  the  aid  of  the  principles. 

(xy  Z  z)  (w  Z  x)  Z  (wy  Z  z), 

(xy  Z  2)  (w  Z  y)  Z  (xw  Z   2), 

(xy  Z  2)  (s  Z  w)  Z  (xy  Z  w), 

(xy  Z  z)  Z  (yx  Z  z). 

§  33.  The  valid  moods  of  the  syllogism, 

x(a,  b)y(b,  c)  Z  z'(ca), 
one  hundred  and  forty-two  in  number,  as  well  as  those  of  the 
syllogisms,  x(a,  b)y'(b,  c)  Z  z'(ca)  and  x'(a,  b)y(b,  c)  Z  z'(ca)t 
which  number  thirty-one  and  twenty-seven  respectively,  may  now 
be  obtained  from  the  results  of  §  32  and  the  forms  of  immediate 
inference  by  the  aid  of  the  additional  principles, 

(xy  Z  z')  Z  (xz  Z  /), 

(xy  Z  z')  Z  (zy  Z  x'). 
The  examples  which  follow  will  be  enough  to  illustrate  the 
method. 

(1)  {y(ba)y(cb)  Z  y(ca)\  Z  {y(ba)y'(ca)  Z  y'(cb)) 
by  (xy  Z  z)  Z  (xz'  Z  y'). 

(2)  \y(ab)y'(cb)  Z  y'(ca)\  U(cb)  Z  y'(cb)) 

Z  {y(ab)t(cb)  Zy'(ca)}, 
by  (xy  Z  z)  (w  Z  y)  Z  (xw  Z  z). 

(3)  \y(ab)t(cb)  Z  y'(ca))  Z  \y(ab)y(ca)  Z  t'(cb)}, 
by  (xy  Z  z')  Z  (xz  Z  y'). 

No  other  valid  moods  of  syllogistic  form  exist,  except  those  that 
have  now  been  enumerated,  as  will  appear  in  the  sequel,  when  all 
of  the  remaining  variants  shall  have  been  declared  untrue. 

Exercises 

1.  From  postulate  (xi)  deduce  six  valid  implications  of  the  form, 
x(a,  b)y'(b,  c)  Z  z'(ca). 

2.  From  postulate  (xi)  deduce  thirty-three  valid  implications  of  the 
form,  x(a,  b)y(b,  c)  Z  z'(ca). 

§  34.  It  will  be  convenient  in  establishing  the  invalid  moods 
of  the  syllogism  to  begin  with  the  form 

x(a,  b)y(b,  c)  Z  z'(ca). 

Any  invalid  mood  under  this  head,  which  contains  an  a-premise 
or  an  a-conclusion,  may  be  shown  to  be  invalid  by  identifying 
terms  in  the  a-form.     Thus: 


Foundations  of  Formal  Logic 


43 


1.  Suppose  a{ba)y(cb)  Z  y'(ca)  were  valid,  and  identify  terms 
in  the  major  premise. 

[a(aa)y(ca)  Z  y'(ca)\  [i  Z  a(aa))  Z  \y(ca)  Z  y'(ca)}, 
by  (xy  Z  z)  (w  Z  x)  Z  (07  Z  0). 

.' .a(ba)y(cb)  Z  7'(ca)  is  invalid. 

2.  Suppose  7(a6)7(c6)  Z  a'(ca)  were  valid  and  identify  terms 
in  the  conclusion. 

(7(06)7(06)  Z  a'(aa)}  {a'(aa)  Z  0}  Z  (7(06)7(06)  Z  0}, 
by  (xy  L  z)  (z  L  vj)  Z.  (xy  Z  w) ; 

{7(06)7(06)  Z  0}   Z  {7(06)  Z  7'(o6)J, 
by  (*}>  Z  o)  Z  (x  Z  /). 

.'. 7(06)7(^6)  Z  a'(ca)  is  invalid. 

Exercise 
Establish  the  invalidity  of  the  thirty-four  moods  of  the  syllogism, 
x(a,  b)y(b,  c)  Z  z'(ca),  which  are  invalid  and  which  contain  an  a-premise 
or  an  a'-conclusion. 

In  order  to  deduce  the  invalid  moods,  which  remain,  eighty  in 
all,  it  will  be  necessary  to  add  eleven  postulates  to  the  ones  already 
set  down.     These  assumptions  are : 


fl(ba)(3(cb) 

0(6o)/3(c6) 

^(6o)/3(c6) 

7(06)7(^6) 

7  (6a)  7  (6c) 

7(6ah(c6) 

7(06)7(^6) 

t  (6a)  7  (6c) 

<(6a)7(6c) 

e(6a)«(c6) 

«(6a)«(c6) 


Z  j8'(ca 

Z  y'(ca 

Z  t(ca 

Z  |8'(ca 

Z  /3'(ca 

Z  y'(ca 

Z  t'(ca 

Z  /3'(ca 

Z  t'(ca 

Z  /3'(ca 

Z  e'(ca 


is  an 
is  an 
is  an 
is  an 
is  an 
is  an 
is  an 
is  an 
is  an 
is  an 
is  an 


invalid 
invalid 
invalid 
invalid 
invalid 
invalid 
invalid 
invalid 
invalid 
invalid 
invalid 


mood, 
mood, 
mood, 
mood, 
mood, 
mood, 
mood, 
mood, 
mood, 
mood, 
mood. 


Exercise 
From  the  postulates  just  set  down  deduce  sixty-nine  other  non-impli- 
cations of  the  same  form,  by  the  aid  of  the  additional  principles. 

(xy  Z  z)'(w  Z  2)  Z  (xy  Z  w)' , 
(xy  Z  z)'(x  Z  w)  Z  (wy  Z  2)', 
(xy  Z  2)'(y  Z  w)  Z  (xw  Z  2)', 

(xy  Z  2')'  Z  (xz  Z  y'Y, 

(xy  Z  z')'  Z  (zy  Z  *')', 

(*y  Z  2)'   Z  (y*  Z  2)'. 


44         Foundations  of  Formal  Logic 

§  35.  All  of  the  invalid  moods  of  the  syllogistic  form,  x(a,  b) 
y(b,  c)  Z  z{ca),  x'(a,  b)y(b,  c)  Z  z'(ca)  and  x(a,  b)y'{b,  c)  Z  z'(ca), 
may  be  deduced  at  once  from  the  results  that  have  now  been 
established.  A  few  examples  will  be  enough  to  illustrate  the 
method. 

U(ba)y(bc)  Z  0'(ca)}'  Z  {0(ca)e(ba)  Z  y'(bc))', 
by  (xy  Z  z')'  Z  (zx  Z  /)'; 

lP(abWcb)  Z  y'(ca)\'  \0{db)  Z  y'(ab)) 
Z  W (ab)e(cb)  Z  y'(ca)}', 
by  (xy  Z  z)'  (x  Z  w)  Z  (wy  Z  2;)'; 

\y'(ab)e(cb)  Z  7'(ca)}'  Z  U(c&)7(ca)  Z  7(06)}', 
by  (z'y  Z  z')'  Z  (yz  Z  *)'; 

\y'(ab)e(cb)  Z  7'(«0}'  Z  |7'(ab)7N  /-  *'(d0}', 
by  (*y  Z  2')'  ^  (**  ^  /)'; 

U(ba)y(bc)  Z  0'(«*)}'  {e(az)  Z  &'(ca)) 
Z  {e(&a)7(M  Z«(Ca)}', 
by  (%7  Z  0)'  (w  Z  z)  Z  (^  Z  m;)'; 

{e(6a)7(6c)  Z  e(ca)}'  J€(ac)  Z  e(ca)\  Z  {7(6c)«(6o)  Z  e(a<r)}', 
by  (xy  Z  2)'  (w  Z  z)  Z  (yx  Z  w)'. 

Exercises 

1.  Show  that  there  exist  no  valid  implications  of  the  form  x'(a,  b) 
y(b,  c)  Z  %(ca)  or  x(a,  b)y'(b,  c)  Z  z(ca),  and  consequently  none  of  the 
form,  x'(a,  b)y'(b,  c)  Z  z(ca)  or  x'(a,  b)y  (b,  c)  Z  z'(ca). 

2.  Show  that  as  a  result  of  a  complete  induction  of  the  moods  in 
question  (a)  a  valid  mood  of  the  syllogism,  whose  premises  and  conclu- 
sion are  all  unprimed  forms  and  one  of  whose  premises  is  of  the  same  form 
as  the  conclusion,  will  remain  valid,  when  the  other  premise  is  put  in  the 
a-form;  and  (b)  a  valid  mood  of  the  syllogism,  whose  premises  are  un- 
primed forms  and  whose  conclusion  is  a  primed  form  and  one  of  whose 
premises  is  of  a  different  form  from  the  conclusion,  will  remain  valid, 
when  the  other  premise  is  put  in  the  a-form. 


CHAPTER   VII 

§  36.  Because  they  possess  similar  properties,  it  is  the  custom 
to  represent  inclusion  and  implication — just  as  the  negation  of  a 
class  and  the  denial  of  a  proposition — by  the  same  sign,  i.  e., 

a  Z  b     =  a  is  included  in  b, 
a  Z  b'    =  a  is  included  in  non-b, 
(a  Z  by  =  a  is  not  included  in  b,  etc. 

"Let  us  take  a  pair  of  contrary  names,  as  man  and  not-man.  It  is 
plain  that  between  them  they  represent  everything  imaginable  or  real,  in 
the  universe.  But  the  contraries  of  common  language  usually  embrace, 
not  the  whole  universe,  but  some  one  general  idea.  Thus,  of  men,  Briton 
and  alien  are  contraries:  every  man  must  be  one  of  the  two,  no  man  can 
be  both.  Not-Briton  and  alien  are  identical  names,  and  so  are  not-alien 
and  Briton.  The  same  may  be  said  of  integer  and  fraction  among  num- 
bers, peer  and  commoner  among  subjects  of  the  realm,  male  and  female 
among  animals,  and  so  on,  In  order  to  express  this,  let  us  say  that  the 
whole  idea  under  consideration  is  the  universe  (meaning  merely  the  whole 
of  which  we  are  considering  parts)  and  let  names  which  have  nothing  in 
common,  but  which  between  them  contain  the  whole  idea  under  consid- 
eration, be  called  contraries  in,  or  with  respect  to,  that  universe.  Thus, 
the  universe  being  mankind,  Briton  and  alien  are  contraries,  as  are  soldier 
and  civilian,  male  and  female,  etc.:  the  universe  being  animal,  man  and 
brute  are  contraries,  etc."     (De  Morgan,  Formal  Logic,  pp.  37-38.) 

Suppose  that  we  should  wish  to  express  the  propositional 
functions,  a,  /3,  7  and  e,  in  the  forms  that  are  ordinarily  employed. 
It  would  then  seem  natural  to  recognize  the  following  identities: 

a(ab)  =  (a  Z  b)  (b  Z  a), 

0(ab)  =  (a  Z  b)'  (b  Z  a)'  (a  Z  b')', 

y(ab)  =  (a  Z  b)  (b  Z  a)', 

e(ab)  =  (a  Z  &'). 

We  might  then  inquire  if  these  representations  verify  all  of  the 
implications  set  down  as  true  in  the  preceding  chapters.  In  order 
to  make  this  verification  complete,  it  would  be  enough  to  deduce 
the  characteristic  postulates  of  our  system  by  means  of  the  trans- 
formations of  the  class  calculus,  which  we  should,  accordingly, 
assume  as  necessity  requires.     Thus  we  should  have : 

(1)  (c  Z  b)  {b  Z  a)  Z  (c  Z  a). 

(2)  (a  Z  b)  (b  Z  c)  Z  (a  Z  c). 

45 


46         Foundations  of  Formal  Logic 

Multiplying  together  both  sides  of  (1)  and  (2)  and  rearranging 
the  factors  conjoined  in  the  antecedent, 

(6  Za)  (a  Z  b)  (c  Z  b)  (Jb  Z  c)  Z  (c  Z  a)  (a  Z  c), 
which  by  definition  would  be  the  same  as 
a(ba)a(cb)  Z  a{ca). 

But  it  would  soon  be  discovered  that  some  of  our  implications 
break  down.    Thus, 

y(ab)e(ab)  Z  o, 
would  be  rendered  by 

(a  Z  b)  (b  Z  a)'  (a  Z  b')  Z  o, 
or  (a  Z  b)  (a  Z  6')  Z  (6  Z  a), 
and  this  implication  is  manifestly  not  true  in  general. 

§  37.  With  the  development  of  the  class  calculus  a  breakdown 
similar  to  the  one  just  noticed  was  pointed  out  among  the  impli- 
cations of  the  classical  logic  and  it  has  been  the  habit  of  logicians 
to  assert  that  the  relation  of  subalternation  and  some  of  the  valid 
moods  of  the  syllogisms  are  fallacious.  This  misapprehension  is 
all  but  universally  shared  by  recent  writers.  Its  removal  may  be 
effected  by  a  solution  similar  to  the  one  which  follows.  (See  also 
Chap.  IX.) 

§  38.  Our  rendering  of  the  four  propositional  functions,  a,  /3,  y 
and  €,  has  been  over-simplified.  Let  us  attach  to  them,  not  the 
meaning  which  they  had  above  and  which  has  proven  insufficient, 
but  the  one  which  follows,  viz., 

a(ab)  =  (a  Z  6)  (6  Z  a), 
p(ab)  =  (a  Z  bY  (6  Z  a)'  (a  Z  &')', 
y{ab)  =  (a  Z  6)  (6  Z  a)'  {A'  +  B), 
t{ab)   =  (a  Z  b')A'B', 
where1  A  =  a  Z  a'  and  B  =  b  Z  b'. 
Consider  the  syllogism  (y*t)i.    We  have 

(1)  (6  Z  a)'  (6  Z  bJA'C  Z  A'C. 

(2)  (a  Z  6)  (b  Z  c')  Z  (a  Z  c'). 

Multiplying  together  both  sides  of  (1)  and  (2)  and  factoring  and 

1  The  assertion  of  c  Z  a'  is  the  same  as  to  assert  that  a  is  a  null-class,  or  a 
class  that  contains  no  objects.  The  class  contradictory  to  the  null-class  is 
called  the  universe.  These  two  may  be  represented  by  the  symbols  o  and  * 
respectively  and  are  defined  by  o  Z  i,  (*'  Z  o)'. 


Foundations  of  Formal  Logic  47 

strengthening  the  antecedent,  remembering  that  (A'  +  B)B'C  Z 

A'C, 

(a  Z  6)  (6  Z  a)'  (A'  +  B)  (b  Z  c')B'C  /.  (a  /.  c')A'C 

or  y(ab)e(cb)  Z  e(ca), 

since  (6  Z  c')  =  (c  Z  6')  and  (a  Z  c')  =  (c  Z  a'). 

It  will  be  easy  to  see  that  the  characteristic  features  of  the 
system,  which  we  have  set  down,  are  now  retained.  Thus, 
a(ab),  0(ab)  and  e(ab)  alone  are  simply  convertible,  a(ab)  becomes 
true  and  the  other  forms  false  when  the  terms  are  identified,  and 
k(a6)w(a6)  =  o.  Two  illustrations  will  furnish  the  student  the 
clue  to  a  verification  of  the  remaining  postulates. 

(a)  (B'  +  ,4)  (C  +  B)  =  B'C  +  AC  4-  AB, 
B'C  +  AC  Z  C, 

AB  Z  A, 
.-.(1)  (Bf  +  A)  (C  +  B)  Z  (C'  +  A). 
(c  Z  b)  (6  Z  a)  Z  (c  Z  a), 
(a  Z  6)'  (c  Z  6)  Z  (a  Z  c)', 
.'.(2)  (6  Z  a)  (a  Z  b)'  (c  Z  6)  (6  Z  c)'  Z  (c  Z  a)  (a  Z  c)'. 
Multiplying  together  both  sides  of  (1)  and  (2), 
(6  Z  a)  (a  Z  6)'  (5'  +  A)(cZ  6)   (6  Z  c)'  (C  +  B)  Z  (c  Z  a) 
(a  Z  c)'  (C  +  A), 
or  y{ba)y{cb)  Z  7(ca). 

(b)  (6  Z  a)  (6  Z  a')  Z  B, 

.\(6  Z  a)  (6  Z  c)  (c  Z  a')  Z  5, 
.'.(6  La)  (b  Z  c)  5'  Z  (c  Z  a')', 
.-.(1)     (6  Z  a)  (a  Z  6)'  (6  Z  c)  (c  Z  &)'£'  Z  (c  Z  a')' 
(A  +  5')  (C  4-  B')  =  AC  +  B'C  +  AB  '+  B\ 
,4(7  4-  B'C  Z  C, 
4B'  Z  A, 
:.AC  4-  B'C  +  AB'  Z  A  +  C. 
.'.(2)  (bL  a)  (a  L  by  (b  L  c)  (c  Z  6)'  (/1C  +  B'C  +  AB')  Z 
(A  +  C). 
Adding  together  both  sides  of  (1)  and  (2), 

(b  /.a)  (a  Z  6)'  (A  4-  B')  (b  Z  c)  (c  Z  6)'  (C  4-  5')  Z  (c  Z  a')' 

+  A  +  C 

or  7 (6a) 7 (6c)  Z  «'(ca). 

§  39.  The  postulates  (p.  43)  may  be  established  empirically  by 
translating  them  into  the  new  mode  of  representation  and  assigning 
appropriate  concrete  meanings  to  the  terms.     Thus, 


48         Foundations  of  Formal  Logic 

y(ba)y(cb)  Z  y'(ca), 
or  y(ba)y(cb)y(ca)  Z  o, 
implies  y(ba)y(cb)  Z  o, 
since  y{ca)  may  be  strengthened  at  once  to  y(ba)y(cb),  and  the 
empirical  untruth  of  this  implication  is  manifest. 

Translating  y(ba)y(cb)y(ca)  Z  o  into  the  new  mode  of  repre- 
sentation, we  should  have: 

(b  Za)(a  Z  b)'  (c  Z  b)  {b  Z  c)'  (c  Z  a)  (a  Z  c)' 
(A  -f  £')  (B  +  CO  (A  +  CO  Z  <?. 
Now,  (c  Z  6)  (6  Z  o)  Z  (c  Z  a) ; 
.'.we  may  omit  the  factor  (c  Z  a). 
And  (6  Zo)(iZ  c)'  Z  (a  Z  c)'; 
.'.we  may  omit  the  factor  (a  Z  c)'. 
And  (A  4-  BO  (B  +  C)  Z  (A  4-  CO ; 
.'.we  may  omit  the  factor  (A  +  C). 
The  result  (6  Z  a)  (a  Z  6)'  (c  Z  6)  (6  Z  c)'  (A  4-  £')  (B  +  CO  Z 
0,  yields, 

(b  Z  a)  (a  Z  b)'  (c  Z  b)  (b  Z  c)71£   Z  o, 
(6  Z  a)  (a  Z  6)'  (c  Z  b)  (b  Z  c)'AC  Z  o, 
(6  Z  a)  (a  Z  6)'  (c  Z  6)  (6  Z  c)'£'C  Z  o. 
Suppose  that  a,  6  and  c  be  particularized  thus : 

Let  a  =  figures  of  less  than  six  sides, 
b  =  figures  of  less  than  five  sides, 
c  =  figures  of  less  than  four  sides. 
Each  proposition  conjoined  in  the  antecedent  of  the  last  of  the 
three  implications  above,  for  extra-logical  reasons,  becomes  true 
and  the  untruth  of  the  implication  as  a  whole  is  manifest. 

§  40.  It  must  be  observed,  however,  that  the  interpretation  of 
our  four  propositional  functions,  which  has  just  been  given,  is  not 
unique.  All  the  implications  of  the  logic  we  have  been  developing 
will  hold,  if  the  following  meaning  were  to  be  assigned  to  the  four 
forms,  viz., 

a{ab)  =  (a  Z  6)  (6  Z  a)  (AB  +  A'B'), 
P(ab)  =  (a  Z  bY  (6  Z  a)'  {(a  Z  60'  +  (A  +  B)), 
y(ab)  =  (a  Z  b)  \{b  Z  a)' {A'  4-  B)  4-  A'B), 
e(a&)   =  (a  Z  b')A'B'. 
Let  one  further  illustration  suffice : 
ABC  +  B'C  Z  A  +  C, 
(b  Z  c)  (c  Z  b)  (a  Z  b)'  Z  (a  Z  c)', 


Foundations  of  Formal  Logic  49 

.-.(1)  (6  Z  c)   (c  Z  b)    [(ABC  +  B'C)   (a  Z  6)'}  Z(cZ  c)' 
(A  +  C). 
AB'C  Z  AC, 
.'.(2)  (b  Z  c)(c  Z  b)AB'C  Z  AC. 
Adding  together  (1)  and  (2), 
(6  Zc)(c  Z  6)  {(ABC  +  B'C)  (a  Z  b)'  +  AB'C'}  Z  (a  Z  c)' 

(A  -f  C)  +  AC 
and  (c  Zb)(b  Z  a)  Z  (c  Z  a) 
.-.(6  Zc)i(aZ  6)'  (A  +  B')  +  AB'\  (c  Z  6)  (6  Z  c)  (BC  +  B'C) 

Z  (c  Z  a)  {(a  Z  c)'(A  +  C)  +  AC'}, 
or  7(6a)o(c6)  Z  tW- 


CHAPTER  VIII 

§41.  In  the  last  chapter  it  was  remarked  that  some  of  the 
implications  of  the  logic  that  we  have  been  developing  would 
break  down,  if  we  were  to  accept  a  traditional  interpretation 
of  our  propositional  functions.  In  that  case  we  should  come 
upon  a  logic  whose  characteristic  postulates  would  stand  in  con- 
tradiction to  those  which  we  have  become  accustomed  to  accept. 

It  is  important  to  observe  that  this  circumstance  does  not 
compel  us  to  conclude  that  one  of  the  two  systems  must  there- 
fore be  false  in  the  absolute  sense  of  that  word.  If  one  system 
is  taken  to  be  true,  the  other  is  by  implication  false,  but  it  may 
well  be  that  the  world  of  common  experience  could  force  neither 
one  of  them  upon  us.  In  point  of  fact  more  than  one  system 
of  inference  is  possible.  A  system  of  logic  whose  characteristic 
postulates  stand  in  contradiction  to  those  of  the  classical  logic 
but  which  finds  its  application  in  precisely  the  same  world,  may 
be  called  appropriately  a  non-Aristotelian  logic. 

§  42.  As  an  illustration  of  method  we  shall  indicate  in  a  set  of 
exercises  the  existence  of  a  system,  whose  underlying  assumptions 
appear  paradoxical  to  ordinary  intuition,  for  this  system  will 
deny  the  truth  of  the  proposition,  all  a  is  all  a,  and  assert  that  the 
proposition,  no  a  is  a,  is  not  untrue  for  all  meanings  of  a. 

In  the  exercises  below  the  following  meaning  is  to  be  attached 
to  the  categorical  forms : 

a(ab)  =  (a  Z  b)  (b  Z  a)A'B' 

/3(o6)  =  (a  Z  b)'  (b  Z  a)'  (a  Z  b')' 

y{ab)  =  (a  Z  6)  (6  Z  a)'  (A'  +  B) 

t(ab)  =  (a  Z  b')  4-  A  4-  B 

Exercises 
(1)  If  k(ab)  and  w(cfc)  can  represent  any  one  of  the  unprimed  forms 
but  can  not  represent  the  same  form,  y(ab)  and  y(ba)  being  considered 
distinct,  show  in  what  cases  k(o6)w(o6)  Z  o. 

»  (2)  Prove  that  the  members  of  the  set  listed  below  are  null-forms 
(see  note  p.  46). 

0(oa')        c(«) 
y(oa)  a(oo) 

y(io)  y(oi) 

50 


Foundations  of  Formal  Logic         51 

(3)  Prove  that  the  members  of  the  following  set  are  one-forms: 

t(oo)  o(u) 

t(oi)  t(oa') 

t{oa)  €(ao) 

(4)  Show  that  the  truth  or  untruth  of  the  following  propositions  is 
contingent  on  the  meaning  of  a: 

a(ia)  e(ai) 

a(aa)  t{ia) 

t(aa)  a(ai) 

(5)  Show  in  what  cases  the  twenty-nine  moods  of  the  ordinary  syllo- 
gism, that  have  been  already  recognized  as  valid,  are  valid  in  the  present 
system. 


CHAPTER  IX 

§  43.  We  shall  now  indicate  the  relation  of  the  system  of  logic 
which  has  been  partially  developed,  to  the  classical  science 
perfected  in  the  Organon  of  Aristotle.1 

The  four  categorical  forms  employed  by  the  traditional  logic 
and  denoted  by  the  letters,  A,  E,  I,  O,  are: 

A(ab)  =  All  a  is  b, 

E(ab)  =  No  a  is  b, 

l(ab)  =  Some  a  is  b, 

O(ab)  =  Some  a  is  not  b, 
the  word  some  expressed  before  the  subject  of  I  and  O  and  under- 
stood before  the  predicate  of  A  and  I,  being  interpreted  to  mean 
some  at  least,  possibly  all. 

The  valid  moods  of  the  Aristotelian  syllogism  are  conveniently  remem- 
bered by  means  of  the  following  mnemonic  lines,  the  vowels  in  each 
separate  word  standing  for  the  mood  in  question: 

Barbara,  Celarent,  Darii,  Ferioque  prioris; 
Cesar  e,  Camestres,  Festino,  Baroko,  secundae; 
Tertia,  Darapti,  Disamis,  Datisi,  Felapton, 
Bokardo,  Ferison,  habet;  Quarta  insuper  addit 
Bratnantip,  Camenes,  Dimaris,  Fesapo,  Fresison. 
This  mnemonic  first  appears  in  the  Summulae    Logicales  of    Petrus 
Hispanus  (afterwards  Pope  John  XXI),  who,  however,  does  not  profess 
to  be  the  author  of  it.     Several  other  versions  are  found  in  later  writers. 
A  Greek  mnemonic  of  the  same  kind  is  inserted  in  editions  of  the  Organon 
preceding  that  of  Pacius.       (From  Mansel's  Aldrich,  Artis  Logicae  Rud- 
imenla,  cap.  Ill,  §  5,  note  z.)     The  moods  not  listed  are  gotten  by  weak- 
ening the  universal  conclusions  to  particular  conclusions  (see  example  5  at 
the  end  of  this  chapter). 

§  44.  The  connection  between  the  propositional  functions, 
A,  E,  I,  O,  and  the  special  forms,  which  have  been  employed  in 
the  text,  may  be  conveniently  assumed  to  be  as  follows : 

A(ab)  =  a(ab)  +  y(ab), 

E(ab)  =  e(o6), 

l{ab)  =  a(ab)  +  P(ab)  +  y(ab)  +  y(ba), 

0(ab)  =   e(ab)  +  8(ab)  -f-  y(ba). 

1  For  a  concise  outline  of  the  traditional  logic  and  good  of  its  kind  the  student 
may  consult  Thomas  Fowler,  Elements  of  Deductive  Logic,  Oxford,  1905.  An 
extended  treatment  of  the  problem  of  this  chapter  is  to  be  found  in  the  writer's 
Letters  on  Logic,  Philadelphia,  1920. 

52 


Foundations  of  Formal  Logic  53 

By  actually  performing  the  indicated  multiplications  and 
allowing  the  product  k(ab)w(a&)  to  drop  out  whenever  it  occurs 
we  should  be  able  to  express  a,  13,  y  and  «  in  the  members  of  the 
set,  A,  E,  I,  O,  thus: 

a(ab)  =  A(ab)A(ba), 
p(ab)  =    l(ab)0(ab)0(ba), 
y(ab)  =  A(ab)0(ba), 
t(ab)  =  E(a6). 

§  45.  In  order  that  the  classical  system  should  hold  true  in  all 
of  its  parts,  certain  characteristic  conditions  must  be  satisfied. 
Besides  the  moods  Barbara  and  Celarent,  we  should  have  to  have : 

1.  Corresponding  to  each  member  of  the  set,  A,  E,  I,  O,  there 
is  another  member  of  the  same  set  which  stands  for  its  contra- 
dictory; 

2.  The  relation  of  subalternation,  A  Z  I,  holds  true; 

3.  The  subject  and  predicate  of  E  and  I  alone  are  simply 
convertible. 

Today  it  is  all  but  universally  taken  for  granted  that  not  all 
of  these  conditions  hold  for  all  meanings  of  the  terms  and  it  is 
usual  to  retain  conditions  (1)  and  (3)  and  to  assert  that  condition 
(2)  is  not  generally  true. 

"The  untruth  of  the  traditional  moods  of  the  syllogism,  by  means  of 
which  from  two  universal  judgments  one  would  deduce  a  particular  judg- 
ment, has  been  recognized  separately  by  Miss  Ladd  (1883),  Schroder, 
Nagy,  Peano,  etc.  It  is  one  of  the  first  and  most  remarkable  results  of 
the  adoption  of  a  logical  ideography."  (Padoa,  La  Logique  Deductive, 
etc.  Revue  de  M^taphysique  et  de  Morale, — T.  20,  n°l,  1912,  p.  67, 
note.)     See  also  the  second  citation  in  small  type  of  §  46  below. 

§46.  Employing  the  notation  of  Chapters  VII  and  VIII,  it 
would  seem  natural  to  recognize  the  following  identities : 

A(ab)  =  (a  Z  6), 
E(ab)  =  (a  Z  b'), 
l(ab)  =  (a  Z  b'Y, 
0(ab)  =  (a  Z  b)'. 

Compare  the  citation  given  below,  wherein  we  have  replaced  a  part  of 
the  author's  notation  by  our  own. 

"The  universal  affirmative:  'All  a  is  b'  is  rendered  by  the  formulas 
(a  Z  b)  =  (a  =  ab)  =  (ab'  =  o)  =  (a'  4-  b  =  i) 
and  the  universal  negative:  '  No  a  is  b'  by  the  formulas 

(a  Z  b')  =  (a  =  ab')  =  (ab  =  o)  =  (a'  +  b'  =  i) 


54         Foundations  of  Formal  Logic 

The  particular  affirmative:  'Some  a  is  b',  being  the  negation  of  the 
universal  negative,  is  rendered  by  the  formulas 

(a  Z  b')'  =  (a  =  ab'Y  =  (ab  =  o)'  =  (a'  +  V  =  »)' 
and  the  particular  negative:   'Some  a  is  not  b',  being  the  negation  of  the 
universal  affirmative  is  rendered  by  the  formulas 

(o  Z  bY  =  (a  =  aft)'  =  (ab'  =  o)'  =  (a'  +  6  =  t)'" 
(Couturat,  L'Algebre  de  la  Logique,  Paris,  1905,  notes,  pp.  82,  83.) 

However,  granting  this  meaning  of  A,  E,  I  and  O,  subalternation 
and  some  of  the  valid  moods  of  the  syllogism  fail.  But  this 
interpretation  of  Aristotle's  four  forms  is  in  no  way  forced  upon 
us.  The  meaning  set  down  below  accords  equally  well  with  the 
usage  of  language  and  with  common  sense.     Let  us  assume : 

A(ab)  =  (a  Z  b)A'B\ 
E(ab)  =  (a  Z  b')  +  A  +  B, 
I(ab)  =  (a  Z  b'YA'B', 
0(ab)  =  (a  Z  b)'  +  A  +  B. 

The  characteristic  features  of  the  Aristotelian  system  may  now 
be  verified.  As  an  illustration  of  the  method  of  verifying  the 
twenty-four  valid  moods  of  the  syllogism,  let  us  select  the  mood 
Darapti: 

"But  with  our  definitions,  All  S  is  P  does  not  imply  Some  S  is]P,  since 
the  first  allows  the  non-existence  of  S  and  the  second  does  not;  thus 
conversion  per  accidens  becomes  invalid,  and  some  of  the  moods  of  the 
syllogism  are  fallacious,  e.  g.  Darapti:  All  M  is  S,  All  M  is  P,  therefore 
Some  S  is  P,  which  fails  if  there  is  no  M."  (Bertrand  Russell,  Introduc- 
tion to  Mathematical  Philosophy,  London,  1919,  p.  164.) 

Thus,  A(ba)  =  (b  Z  a)B'A' 
A(bc)  =  (6  Z  c)B'C 


l{ca)  =  (c  Z  a'yCA' 
We  have : 

(6  Z  a)  (a  Z  c')  Z  (b  Z  c') 
(b  Z  c)  (c  Z  b')  Z  (b  Z  b') 
Accordingly, 

(6  Z  a)  (b  Z  c)  (a  Z  c')  Z  (b  Z  b') 
since  (6  Z  c')  =  (c  Z  b') 
and 

(6  Z  a)  (b  Z  c)B'  Z  (c  Z  a')' 
since  (a  Z  c')  =  (c  Z  a') 


Foundations  of  Formal  Logic         55 

/.(6  Z  a)  (6  Z  c)A'B'C  Z  (c  Z  oO'A'C 

or  {(6  Z  o)B'A'J  {(6  Z  e)B'C'|  Z  }(c  Z  a'yC'A'\ 

which  by  definition  is  the  same  as 

A(ba)A(bc)  Z  I(ca). 

Exercises 

(1)  Verify  the  three  conditions  equivalent  to  those  of  $  44,  viz., 

1.  A(ab)0(ab)  Z  o 
E(ab)I(ab)  Z  o 
A'(ab)0'(ab)Z  o 
E'{ab)V(ab)  Z  o 

2.  A(ab)E(ab)   Z  o 

3.  E(aft)I(*a)    Z  o 

We  might  easily,  had  we  wished,  have  interpreted  A,  E,  I  and  O  in  §  45 
so  as  to  satisfy  the  additional  conditions  A'(aa)  Z  o,  E(aa)  Z  o  (see 
Exercise  3). 

(2)  Verify  the  twenty-four  valid  moods  of  the  Aristotelian  syllogism 
(see  the  paragraph  in  small  type  of  §  43). 

(3)  Show  that  the  truth  or  untruth  of  the  following  propositions  is 
contingent  on  the  meaning  of  a: 

A(aa)         E(ao)         A(ai) 

(4)  If  affirmative  forms  are  those,  which  become  false  when  the  terms 
are  made  contradictory,  and  negative  forms  are  those  which  become  true 
under  the  same  circumstances,  show  that  A(ab)  and  I(ab)  are  affirmative 
and  that  E(ab)  and  O(ab)  are  negative  forms. 

(5)  If  X(ab)  Z  Y(fli)  and  { Y(ab)  Z  X(ab)  \ ',  then  X  (ab)  is  said  to  be 
universal  and  Y(ab)  is  said  to  be  particular.  Show  that  A(ab)  and  E(ab) 
are  universal  and  that  I(ab)  and  0(ab)  are  particular  forms. 

(6)  If  affirmative  forms  are  those  that  become  true  when  the  terms  have 
been  identified  and  negative  forms  are  those  that  become  false  under  the 
same  circumstances,  show  that  A(ab)  and  l(ab)  are  affirmative  and  that 
E(ab)  and  0(ab)  are  negative  forms  upon  the  following  interpretation 
of  A,  E,  I  andO: 

A(ab)=  (aZ  b)(A'+B), 
E(ab)=  (aZ  6')(--l,+  ^.), 
l(ab)  =  (aZ  b')'+AB'u 
0(ab)  =(oZb)'+  AB', 
where  B,  =  b'  Z  b. 

(7)  Assuming  the  equalities  of  Exercise  6,  show  that  the  relation  of 
obversion,  A(ab)  =  E(ab),  holds  true. 

(8)  Assuming  the  equalities  of  Exercise  6,  show  that  conditions  1  and  2 
of  §  44  are  satisfied. 

(9)  By  means  of  the  same  equalities  verify  the  two  moods  Barbara 
and  Celarent. 

(10)  From  the  two  moods  of  Exercise  9  deduce  ten  other  valid  moods 
of  the  Aristotelian  syllogism  by  means  of  the  principles  of  Chapter  VI. 


56         Foundations  of  Formal  Logic 

(11)  Interpreting  A,  E,  I  and  0  as  in  exercises  (l)-(5),  establish  the 
invalidity  of  the  following  moods  of  the  syllogism, 

AAO,,  EEI,,  AAO< 

AEI1(  OAO„ 

AEOi,  AAA4, 

either  by  allowing  the  terms  to  take  on  the  limiting  values  0  and  i  in 
appropriate  ways,  or  "empirically"  by  assigning  to  them  other  concrete 
meanings  as  in  §39,  Chapter  VII. 

(12)  From  the  seven  moods  listed  above  derive  the  remaining  two 
hundred  and  twenty-five  invalid  moods  by  means  of  the  principles  of 
§34,  Chapter  VI. 


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